-They didn't make a move all year, they're out, you're in. Can go in whatever direction you like with the team.
-Trade deadline is passed, will open up again shortly after the season is over. And yes, can trade draft picks, or players for picks, in the offseason.
-Keeper picks can be made whenever. Can go in and put them in now, but don't become effective until right before the draft, so mid-August give or take. If you're smart, you'll at least put some tenative keeper picks in at the end of the season, so if you forget and miss the deadline, you still get the guys you want. Every year, someone forgets to set them, so set it at the end of the year, and then only have to remember to check/update them.
Welcome aboard, hope you suck as badly as you have in the TNZ league