1: "Tank" is a common DPS term for the character whose job it is to take hits for the rest of the party. They're typically the de facto leader, in charge of pulling groups of monsters and holding their attention, both because they take less damage than anyone else and because a single target is easier for the healer to heal. DPS stands for "Damage per Second", but has come to mean a Damage Dealer class, the folks who care about DPS. Their job is to do as much damage as they can, though this is slightly complicated by the need to not grab the attention of the monsters, and often times there are priority targets to focus on as well.

2: A free company is a collection of characters banded together as a group. They're like guilds in other games, and your FC is typically who you do endgame events with. A closely related thing is a linkshell, which is also a collection of characters. The difference is that a FC is progress-oriented, with group housing, group chest, group buffs, basically all pulling together for a common goal, while a LS is social-oriented, only giving you a shared chat channel (But you can have up to 8 linkshells but only one free company). An example is that you'd have your free company that you run coil and do moneymaking activities with, while having a roleplay linkshell and a linkshell for fans of a particular anime.

3: You will start doing group activities around level 15. The main quest will lead you there, so please look forward to it.