I feel like this was a strong game for me, up until (what I thought afterward) was the questionable move 31 . . . Qe5.  I failed to see before moving that his response 32. d4 put my queen in an awkward position where she could no longer defend against both the mate at g7 and the rook at h5.  Low on options, I chose the suicide check 32 . . . Rxf3+.  Obviously, if 33. Kg1 then 33. ...  Qe1 #.  But as far as I can tell, neither of his other options (32. Rf2 or 32. Qxf2) were bad.  Did I miss something or did he?

33. Rf2?? would lose very badly, but I think your right, 33. Qxf3 and it looks to me white is slightly better.

33. Rf2 leads to 33... Rxf2+ 34. Kxf2 Qxh2+ and black is going to net white's queen as well.

A. In case of the white move 33.Qxf3,

   33. Qxf3 Qf5 34. Qxf5 Rxf5+ 35. Kg1

    The white holds the advantage of 1 pawn.

    This is the white's best move.

B. In case of the white move 33.Rf2,

   33. Rf2 Rxf2+ 34. Kxf2 Qxh2+

    The black wins.

C. In case of the white move 33.Kg1

    33. Kg1  Qe1#. 

Genenally, the black is better.

The white's best move is 33.Qxf3.

If the white had moved to Qxf3, the white could have won this game.

 Not at all. Black will have no problem grabbing some pawns with the move and the exposed white King.