No, you can have your gamertag on one of every team on gamebattles. But you cant have your gamertag on the same two ladders, (You can't have 2 MW2 Doubles teams with the same gamertag), Understand?
Yup, any teams you want to be on and any games, as long as you don't have 2 of the same game/ladder team and use the same gamertag. Only one team per gamertag. (i.e - If you want to make a cod4 doubles team with your friend, you put your gamertag in, then you want to make a doubles team for the same game with a diff friend, you can't, one team per gt.)
To level up, you have to play gamebattles matches. To join a team, say you have xbox 360 and want to make a Call Of Duty 4 Singles Team, you go to xbox 360 - call of duty 4 modern warfare - join ladder - singles ladder - make your team. Then you go to the match finder, and accept or post matches. You then play the matches and whoever wins/loses reports the score. That's how you level up. I suggest you reading the rules of Gamebattles before anything. Also to report or look at a match, you hit the magnifying glass next to the match. Hope that was helpful.
Yup, it's called disbanding =)