Yes, you could sometimes sacrifice the queen for a minor piece, it can be a game-changer.
Would you ever give up a rook for a minor piece?

It all depends on the position. If the bishop is dominant and has no opposing bishop, and is stopping you from doing things like castling and doubling rooks, then taking it could be a real game changer. Who doesn't like an exciting game? But you have to make sure you have adequate compensation, a pawn or two. And don't trade pieces off.
There is some discussion of this sort of thing in the 2014, New in Chess book, Improve Your Chess Pattern Recognition by Arthur van de Oudeweetering. Chess Lecture sells a DVD, The Exchange Sacrifice on c3 in the Sicilian by GM Smith. Apparently Scheveningen and Dragon variations are discussed.
A 2004 Carlsen-Dolmatov game began 1 Nf3 f5 2 d3 d6 3.e4 e5 4 Nc3 Nc6 5 ef Bxf5 6 d4 Nxd4 7 Nxd4 ed 8 Qxd4 Nf6 9 Bc4 c6 10 Bg5 b5 11 Bb3 Be7 12 O-O-O Qd7 13 Rhe1 Kd8 14 Rxe7

Only to stop a checkmate or help me in a checkmate.
Yeah, never for positional advantage. NEVER!

Yes of course. Exchange sacrifices are common. An example is taking the c3 knight in the Dragon, but I would sacrifice my Queen for a pawn,if it helped me win.
Go over Topalov's games, he is a master at exchange sacrafice. It is a pretty common sac, especially if you can pick up a p some where.

Only to stop a checkmate or help me in a checkmate.
Yeah, never for positional advantage. NEVER!
http://www.chess.com/livechess/game?id=1280919660 Move 20, you need to enhance your positional understandings.
Nice game, but you missed mate in 1 at 33. ...Qxe4#

Only to stop a checkmate or help me in a checkmate.
Yeah, never for positional advantage. NEVER!
http://www.chess.com/livechess/game?id=1280919660 Move 20, you need to enhance your positional understandings.
You could also improve on understanding sarcasm.

I've seen it done in a book before. Yesterday I was playing in the park and the other bishop was being a pain. First it prevented me from castling and then it prevented me from doubling up my rooks, and then it took part in the checkmate. I was thinking about taking it out with my rook but didn't wanna fall behind on material. Would it make a difference to you if you got a pawn with it