Today I reached a rook and seven pawn vs. rook and six pawn endgame, which I don't thin I have ever done before. Unfortunately, shortly after reaching it, my opponent essentially gave up (he spend most of his clock on one move for no reason, then made a series of bad moves before waiting for his clock to time out). But had he played on, I don't think it would be so easy for me to win.

From the first position, how should white have played for a draw? Did he have any drawing chances?

I think black should win this because of the passed h pawn, but I'm not 100% sure.

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That must have been a long endgame. Who won?

I don't see how White has any chances from the initial position.  The passed pawns on the kingside + the control of the only fully open (d) file look overwhelming.

Actually I think g4 would have been a mistake for black though -- the only thing that could make this easier on White would be if Black doubled his g-pawns.  h4 followed by Kh5 and THEN g4 would be killing.