I have lots of questions regarding this mode; thank you for any help!
The usual team for TT is a fighter, a mage, and a jungler. The jungler can be any kind of role. The mage usually goes bot, and the fighter top, but this is more of an unwritten rule, and less of a developed strategy set in stone. 5 is rare, but it does happen. It used to happen more, but it's proven to not be that useful most of the time. Now, more than ever, with the jungle being harder, you rarely see people risk anything to set them back even further. You'll almost always see a Lee Sin, he's just a big of a problem in 3s as he is in 5s. He loses his ability to ward hop, but he can Q to nearly 1/3 of the map due to the placement of the monsters and lanes. Champions should either be powerful early game, or late game. Mid game champs like Vlad and Shen are very bad on that map, because they never have a time to be effective. Pantheon, Darius, or Renekton would be a good example of an early game AD pick. They have powerful early games, and could start building tanky the longer the game lasts to avoid falling off. Mages are often Annie, Syndra, or Cass, due to how quickly they become a threat. I personally love late game champs on that map like Anivia, but you have to be prepared for the hyper aggressive early game that many players will try to do (starting with all flat runes and two Doran's Rings is pretty common, although it falls off really hard if it doesn't work).
Above all, the most important thing to keep in mind is that glass/pure tank builds never work. All champs should have a balance of damage and tankiness. You can go full damage early on If you want, but if the game lasts longer than 15-20 mins, you'll begin to die in every fight without being able to do anything unless you get a really good engage.
Personal Examples
Typical Anivia Build: 
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Typical Maokai Full Build: 
I have another question to add to this~ Why are the minions and towers in 3v3 the old minion and tower models?
Im sure you have already gathered some useful information from previous posts, but I will add my 2 cents for Twisted Treeline, since its a game type I play often. I pay close attention to the challenger teams in 3s and their comps, and have looked a little into the EU challenger TT. It seems a very popular to run an AP ranged carry with a classic AD carry and Support (Braum and Leona being good examples). The ADC and Sup actually go top lane, because bottom lane is the safest and easiest to solo farm regardless of 1v1 or 1v2 lane, if your playing Ahri, Liss, Annie, LB, etc, then solo bot should be easy for you. The top lane should dominate with heavy ADC dmg coming with Sup zoning and CC. Popular ADCs are ones with good mobility and or high burst. Kalista, Graves, Corki, Ez.
The Tank/Mage/Jungler approach is also popular and can work fine so long as the team understand thats map positioning changes frequently on TT and moving as a unit will win the game. Having a solo lane and a duo lane makes moving as a unit much easier. If the other team sends a solo laner top into you adc/sup thats usually an easy first blood.
