12... Qe7+ 13. Kd1 Bg4+ 14. f3 Bg7 wins the queen.
19... Qxc2 seemed fine, perhaps even the best move. Everything else gives up the bishop and you'll lose some pawns in the process.
As for move 22... It was too late to save the queen, 20... Kg6 or Kf8 would've been best, so that the bishop and king can both take part in the king's defense.
Not too good at endgames, so I'll let someone else handle those.
Sigh. I just drew a depressing game where I was up by a bishop and two pawns but screwed up with a few silly moves. I was just wondering though, could I have salvaged the win even towards the end? I'm talking about around move 39 or so.
To summarize the main questions in the comments...
Move 12: Did I have a way to trap the queen?
Move 19: Was taking the rook an unsound move?
Move 22: Did I have a way to keep my queen?
Move 30: What is the general plan in endgames versus a lone queen? Was I correct in trying to defend all my pieces, or would a more active plan be better?
Move 39: Was the result already fixed as a draw after this point, or was there some plan that could have rescued the win?
Move 55: Was there any reasonable plan where I could give up material for counterplay, or was this just helpless flailing? (Okay, to be honest I think the answer to this question is very likely the latter, but well, one can hope).
EDIT: Forgot to mention, also, in the final position, did I have any chances? I went for a draw by repetition because I couldn't see anything winning, but given that it was a queen+pawn vs two rooks endgame I think there could have been some interesting imbalances to make use of.