Warning: People say TapJoy is a notoriously bad company who will fight you tooth and nail not to pay out the reward if something goes wrong. I was pretty lucky here, but people hate them for a reason.

Intro

I thought it'd be useful to compile a list of games that people have tried through TapJoy and what kind of strategies people used to most efficiently get to their goal.
I'm going to be straight up - unless you really enjoy the side game you're playing, you're going to feel like it's a waste of time and not worth the gems. I'd only really recommend these TapJoy games to people who are F2P or pact players, or for dolphins with some extra time on their hands. While doing these offers it's easy to suffer from a sort of mobile game burnout. Even playing Alchemist Code was exhausting me.
There is a time limit for some of these challenges (it differs per game, so be mindful of it when you start the challenge), you can only pick one challenge per game, and it has to be a new account in the game. I did not spend any money on any of these games. I spent as much time throughout the day as I could on each game, which was probably about 2 hours a day.
For all of these games, my strategies reflect trying to achieve the challenge ASAP. If you end up liking the game and want to keep playing, some of these strategies may not be optimal, because they require using resources that you'd otherwise want to save.
I would recommend keeping the game installed until after you get your pay-out, and try not to fall below the challenge threshold until you get your payout.
Also, I did notice that I got the same offers on my iPhone after I already completed the offer on an Android device. So these may be repeatable across platforms, but I haven't tested this, and knowing Gumi, there's a chance they'll think you're cheating.
Here is another great thread that goes over TapJoy offers.
Finally, the jewel rewards listed here are based on what can be obtained on an Android device in the U.S. Other countries or platforms may provide differing rewards, so if you find that the reward is different than what is listed, just weigh your reward amount versus my time spent to see if it's worth it for you.

Final Fantasy XV: A New Empire

Challenge Options

10K power (shortest, 579 gems), level 10 citadel (medium, 850 gems), 100K power (longest, 1428 gems)
I picked 100K power

Jewels Obtained

1428

Time Spent

5 days

Jewels Obtained/Time Spent Ratio (Higher is Better)

285.6

Genre

Kingdom builder

Opinion

This is actually a pretty efficient offer. Unlike Lords Mobile, the limiting factor on building up your empire is resources, not time. This makes it much faster to progress. This was the first TapJoy offer I picked up, and maybe because of that, I didn't mind it too much. I've played these sort of kingdom builder games in the past, so it didn't take long for me to catch on.
One thing I hated is that the game is VERY in your face about in-app purchases. I minimize and maximize games a lot; and every time you go back into this game, there's an IAP offer shoved in your face. It's annoying, but a small price to pay.

Strategy

I went for the 100K option, so my strategy is geared towards that. You're going to want to make the following buildings:

• 11 quarries
• 4 farms
• 6 mines
• 5 banks
• 4 energy extractors
• 8 barracks
• 1 hospital ward

Quarries are the most important here, because it's the most used resource. You're going to want to build the buildings above, then level up your citadel whenever possible. This lets you upgrade your buildings to match your citadel level. Outside of that, you can follow the buildings that the game recommends, as those will pay out rewards.
You're going to use your training grounds to make as many siege weapons as possible, as these units hold the most resources. When you open the map view, you'll see resource nodes throughout. You can send your troops to these nodes, and they'll slowly gather that resource type. You'll also see monsters on the map. It's important to attack these monsters, as they'll give you additional resources. Always attack once at a time - don't do the high energy attack.
Keep an eye out for the events going on as you play. There are events for building, researching, training, and attacking monsters. If possible, spend your resources on whatever activity type is going on for that event. You only get points when you finish the building/research/troops, so plan accordingly. These events can pay out with useful resources, speed-ups, or shields.
Joining an active guild is very important. When you're building or researching you can ask for help - everybody who gives you help shortens the build/research time by a minute. Further, when asking for help, and when giving help to others in your guild, you get guild coins that you may use for useful items like speed-ups or resources. Getting into a good guild right from the beginning probably won't happen. The "open" guilds, that allow anybody to apply, are usually pretty inactive; but you may need to start off here. After you've built up a little bit, and if the guild isn't active, leave and rejoin a closed group. You want a guild with active members who hit that "ask for help" button often, since that's where you get your guild coins.
One of your most important buildings, in terms of power gain, is your wall. Do not build the "reinforced wall" unit inside of the wall building. Instead, wait until you've researched traps, and then build nothing but traps. Traps give 4 power each, whereas the walls only give 1.
Just keep upgrading your resource buildings and building siege units and traps as you slowly upgrade your citadel. When you get to citadel level 9, you should have enough shield items to keep you shielded permanently from now until you hit 100K. So make sure you always have a shield active and just rush for 100K by building a lot of units. Forget about research. You had also may as well start using all of your speed-up items to reach that goal ASAP.
After you get your payout, if you had a good guild, do them a favor and donate your resources.

Lords Mobile

Challenge Options

Gather 50K timber on the map (136 gems), reach player level 25 (442 gems), castle level 11 (595 gems), castle level 12 (769 gems), castle level 14 (1530 gems)
I picked level 14 castle

Jewels Obtained

1530

Time Spent

12 days

Jewels Obtained/Time Spent Ratio (Higher is Better)

127.5

Genre

Kingdom builder

Opinion

I actually liked the gameplay in this game better than FFXV: A New Empire. It has a couple of extra modes that make the game more enjoyable, though it's otherwise the same game with a medieval fantasy theme... but there is one other big difference... BUILDING THINGS TAKES FOREVER. For example, building up my citadel required me to level up my wall, which required me to level up my forge, which required me to level up my mine... and each of those start taking at least 15 hours towards the end.

Strategy

I went for the level 14 citadel option, so my strategy is geared towards that. In this game, you want to build up a bunch of farms. Most of your resource income will come from the resource nodes that you see on the map, and not from actual resource production. In your resource build slots, build one timber mill, one mine, one quarry, and then nothing but farms.
You'll unlock the rest of your base by beating some NPC enemies. As you do that, you're ultimately going to want just one barracks, three infirmaries, and the rest Manors. Any additional resource nodes you unlock should be more farms.
Joining an active guild is very important. When you're building or researching you can ask for help - everybody who gives you help shortens the build/research time by a minute. Further, when asking for help, and when giving help to others in your guild, you get guild coins that you may use for useful items like speed-ups. Getting into a good guild right from the beginning probably won't happen. The "open" guilds, that allow anybody to apply, are usually pretty inactive; but you may need to start off here. After you've built up a little bit, and if the guild isn't active, leave and rejoin a closed group. You want a guild with active members who hit that "ask for help" button often, since that's where you get your guild coins.
You're going to be in a bit of a time crunch in this game, because getting to level 14 citadel takes a long time. Ignore the suggested buildings. Instead, look at what your citadel needs for the next level, and go for that. Then just keep doing that. The ONLY exception is that you want a level 9 Academy ASAP. This lets you get tier 2 troops, which is a necessity between citadel levels 9-14. Anyway, as you get towards the end, the buildings start taking a VERY long time. This is when you're going to start using those speed-up items.
Do not claim your "Turf Quest" rewards until you get a "+25% Player XP" boost item. I forget what got me that; but once I got that I used it and claimed all of my Turf Quests in order to get a ton of level ups. When you level up your hero, beeline towards Construction I, max it out, and then go for Construction II, and max it out. After that, go for Research I and Research II. Other than that, it doesn't really matter.
For units, I'd go with a fairly even mix, mostly to help you beat the NPC enemies in your turf. Since resources are so abundant, don't worry too much about the fact that siege engines and infantry carry more than archers or cavalry.
Go through the little dungeons to keep your heroes leveled up and equipped, and then use the heroes to attack monsters on the map. Always attack once at a time - don't do the high energy attack. Monsters can drop ever valuable gems or speed-ups.
After you get your payout, if you had a good guild, do them a favor and donate your resources.
:Edit: XXAlastairXX pointed out that if you don't mind spending $9.00: "buy the 99 cent bundles 4 times and one of the 5 dollars ones. I forgot which one but buying all those in Lords Mobile made getting the castle to level 14 so easy. You could probably do it in 30 minutes."

Empire and Puzzles

Challenge Options

Beat level 51 (province level 7)

Jewels Obtained

300

Time Spent

4 days

Jewels Obtained/Time Spent Ratio (Higher is Better)

75

Genre

Similar to Bejeweled/Candy Crush, but the jewel combinations will hit enemy units, who are attacking you. There's a gacha system to get better units. There's also a sort of base-building mechanic, but you don't really attack other people/get attacked in the same way as in a kingdom builder game.

Opinion

I pretty much only picked this game up because my Fiancée plays it. I probably enjoyed it more than I otherwise would have because of it. It's pretty easy until you get to province 7, and then you'll need items to progress, but that's not a problem. If you like Bejeweled or Candy Crush, go for it - but otherwise, the small payout isn't worth the time investment.
I do like that the game has an auto-battle mechanic that you can use when fighting normal enemies or when fighting titans.

Strategy

The game gives you a starting roster who will be decent enough for the most part. I kept the Fire element healer, but otherwise swapped out all my other 1 star characters. In the end I had Bane (3* light), Tyrum (3* dark; I got lucky with a daily pull, your 2* Layla should be fine), Needler (2* green), Ragnhild (2* blue), and Sharan (1* red). As you turn in jewel combinations during battles, heroes of the color of that combination get mana, which they may use to cast special abilities.
You can get more characters by beating levels, or, more commonly, by training them through the training camp. Before you get to level 4 training camp, you'll constantly hit the manpower cap. Don't worry about this too much. At level 4 training camp you can basically queue up all of your manpower into the camp, so that it will slowly churn out common/uncommon recruits for you throughout the day.
You level up your heroes by feeding other heroes into them. If you feed them units of the same element, it provides additional experience. There's also an unfortunate chance mechanic here: each time you feed units to your hero, there is a chance that their skill will level up. If you feed a copy of the unit to itself, that's +25%, and then each other unit will give +2% for the same element, or +1% for an off-element.
For this challenge, the Forge building is probably more important than it otherwise would be. You want to get your Forge to level 4 for the arrow item; level 5 is also useful for the better healing potion. As we get towards the end of this challenge, these items will be absolutely essential to completing the stages.
You should be able to beat province 6 with fairly minimal use of items, except for maybe some minor healing potions. I feel like at province 7, level 5, things started getting harder. This is when I started needing to use the arrows at each group of enemies. This made the stages much easier.
Joining a good guild in this game isn't as important as in the kingdom building games. When you're in a guild, there are raid monsters called titans that show up. If the guild kills them, you get some rewards. When your guild kills five of them, you get a bonus treasure chest. That's nice, but not a necessity. So joining and staying with an open guild should be fine enough.

Legendary: Game of Heroes

Challenge Options

Reach rank 20

Jewels Obtained

500

Time Spent

1 day* (I got my payout for reaching rank 14, even though the challenge said to reach rank 20)

Jewels Obtained/Time Spent Ratio (Higher is Better)

500

Genre

Similar to Bejeweled/Candy Crush, but the jewel combinations will hit enemy units, who are attacking you. There's a gacha system to get better units.

Opinion

This is similar to Empire and Puzzles. The art style is better, but I think it lacks some of the charm that Empire and Puzzles has. There's no story in the game at all, and there is no base building. It all feels really bland, like you're just aimlessly going through the motions with no real goal. The jewel combining aspect is different from Empire and Puzzles for the better - you can combine jewels diagonally, and when you combine jewels you have a period of time where you can either make additional combinations, or move one jewel to an adjacent location. Your characters then attack randomly, rather than the jewels going out in front of you like in Empire and Puzzles. This is nice since you don't ever feel like you're stuck in combat.

Strategy

The game will give you a choice for your starting character. I went with the blue one, but I doubt it matters much. You'll be able to go through the initial levels very easily. The color and type of tower will correspond with the enemies that you will fight. The sort of normal stone towers are green enemies - other than that, it's pretty self-explanatory, red towers are red, blue are blue, etc. So you'll want to use characters of the opposite element type (red > green, green > blue, blue > red, dark and light beat each other), if possible. After a little while, it'll allow you to join a guild.
Joining a decent guild is a pretty big one here. When you're in a guild, you can use guildmate's characters, which then go on a 24 hour cooldown. So getting into a guild where a lot of people have good units for you to use is pretty important, since their characters will basically solo any levels you come across. A lot of times I'll use a roster full of characters opposite of the map that I'm playing, and then a mercenary of the same color as the tower, or a light or dark mercenary. The mercenary should be able to solo anything, unless maybe it's their opposing element. So by having a mercenary that doesn't match your team's color, you can make use of two different jewel colors rather than having your entire team be one color and have most gems being useless to you.
Just keep progressing through the story. Open up the 500 gem hero promo packs whenever you get the gems for it - chances are, whatever you get here will be better than the common/uncommons you've been picking up along the way. I only bother upgrading the heroes I get from these packs, as well as the starting hero that I picked.
I got the reward when I got to rank 14... not sure if that is a bug that affected just me or if everybody would get that. I can't imagine my strategy would have changed much between ranks 14-20... just keep chugging along through the campaign.

Spin & Win

Challenge Options

Reach level 8 (80 gems), reach level 18 (300 gems)

Jewels Obtained

300

Time Spent

4 days

Jewels Obtained/Time Spent Ratio (Higher is Better)

75

Genre

Slots Casino

Opinion

I'm not a fan of casinos, so virtual casinos are especially "meh". The game does have an auto-play mechanic, but you need to be semi mindful of the game because you have to click on the screen occasionally for timed events. I really didn't like this game. It's basically just hogging up your phone's time for what seems like a very unneccesary reason. This one is really going to make you wonder, "Why am I doing this for 300 gems?"

Strategy

While you're playing, these little balloons will sometimes appear from players playing at your same row of slots. That said, I like to sit at a row with two or three other players on Emerald 7 (at MGM) or Forbidden 7 (at Mirage), then auto-bet at the lowest bet amount (though betting more does give more XP per spin). I have my volume turned up and my phone next to me to hear the little balloon appear, then you have to click on it above your avatar's head. This will give either chips, experience, or loyalty points (which can be claimed for real life casino trips, but meh). Occasionally a little free spin mini-game will appear if someone in your row gets three "Free Spins" on their slot machine. Those games can pay out a lot. As soon as you unlock the Mirage casino, go there. One of the Free Spin games that you can get there is a double XP boost.
You're going to need to be very mindful of the free 100k chips that come in every two hours. Getting these are going to be vital to completing the challenge in time. You do get something like 400k once a day, but if you're not snagging the two hour chips, I don't think you'll be able to do this in 7 days.
Since this game requires that you leave it running to earn progress, I would combine it with another game allows you to make progress while idle (so basically any other game reviewed here).

Idle Heroes

Challenge Options

Reach level 40 (216), Get 2 5* heroes (239), Enhance Power to 10,000 (199)
I picked reach level 40

Jewels Obtained

216

Time Spent

4 days

Jewels Obtained/Time Spent Ratio (Higher is Better)

75

Genre

Gacha, idle turn-based RPG

Opinion

I actually kind of enjoyed this one. It was a breath of fresh air after kingdom builders, puzzle games, and a casino game.

Strategy

I wouldn't recommend picking the "get two 5* heroes" challenge, as that is going to be luck-based. I dunno about you, but for a difference of 23 gems, I'd have the guaranteed "reach level 40" challenge. I had no five star heroes by the time I got to level 40.
Anyway, the game's tutorial pretty much tells you what you need to know. Just keep upgrading your team of heroes and auto-equipping them. I wouldn't bother spending resources on heroes that are 3* or less unless you got nothing but 3* heroes from the free 3 heroic summon scrolls you get early on. I would recommend getting as far as you can by playing the game manually. When you've hit a point where you're out of upgrade glue or gold, and you still can't beat the level you're on, then just come back to the game in a few hours or so. You can then press the green "Get" button on the level that you're at. That should give you enough materials to make it a couple more levels in - then just repeat the process. Switch out your 3* heroes for 4* heroes or better as you get them.

[Web Game] Who's That Pokemon?

Challenge Options

n/a

Jewels Obtained

0 (should have been 22)

Time Spent

About 30 minutes

Genre

Web quiz

Opinion

This was a "Who's That Pokemon?" web quiz with 40 questions. The page was absolutely riddled with ads, and then you'd have to click next twice to get to the next question. It was an unpleasant experience, and even when I was expecting the payout, I was thinking to myself that it wasn't worth it.

Result

The quiz never paid out. I got to the last question and then there was no next or finish option. So I just had to kind of close the browser. I didn't reach to TapJoy about it. I'm not sure if they would have even been helpful if I had.
As a result, I'd recommend avoiding any quizzes or web-based offers.

Videos

Challenge Options

n/a

Jewels Obtained

1

Time Spent

About a minute

Genre

Video

Opinion

These are a really easy, guaranteed way to get 1 gem... but like... it's 1 gem. I'd only do these if you're 100% F2P, and even then it's probably not worth your time.

Result

These always pay out for me immediately.