OK, so black wins on time here, but if this game were played out, would white or black win? 

I played black here.  I think my game was pretty solid until move 32.  Was it an error to push the pawn?

On move 37, I blocked the queen advance through the open lane...should I have instead checked with my queen?

On move 39, I take a pawn...should I have instead pushed the king out earlier?

Thank you for your comments.

You appear to have accidentally posted a diagram rather than a game. Sorry.

Thanks tonydal....that is a nice move!

I like black's passed pawns.

You should trade queens and run those up the board (assuming you didn't lose on time)

U should have checked on move 37 you would have either had checkmate in a few moves depending on his actions or you would have had great position with a large pawn advantage in two moves if he had blocked with the queen. Move 39 would have also been checkmate like the person said below me. 

tonydal's suggestion of 43.Qh2 in the final position is very interesting. As to who's winning after that. Hmmm not sure what happens. But black seems to have an easy draw at hand if he wants it with 43...Ne4+.

Now, after  44. Bxe4 Qxe4 white lost his attack and black's queenside pawns will decide the game.

or 44.Kxg4 Nf6 and I can't find anything better for white than a repetition:

going to the h file with the king just allows black to chop of the d5 pawn for nothing while white's attack is blocked.

44.Kf3 Qxd5 wins the white bishop.

44.Kg3 Qb3+ and either the queens are exchanged (if white king goes to second rank) or white king goes to h file, blocks the queen and black plays ...Qxd5.

So, is it a draw? does black have anything better? or does white have anything other than 43.Qh2.

I have no idea:)