Pawn Structure: What is Black's best move in this position?
Material is even but Black would be happy to trade off all the pieces to reach the following king and pawn ending.
Black is effectively a pawn ahead here due to White's tripled c-pawns. Black will be able to create a passed pawn with his extra pawn on the king side while White won't be able to do so on the queen side. 15...Qxd1+ 16.Kxd1 0-0-0+ would have been very pleasant for Black.
Pieces Before Pawns: A basic endgame principle is that a player should improve the position of his pieces before he advancing his pawns unless the position involves a pure pawn race to see who queens first.
Black played 34...c6 here. It wasn't a bad move, but he should have been looking for a way to exploit the superiority of his rook over the White knight. 34...Re1 intending 35...Ra1 would have shortened White's resistance.
Confining Over Attacking: You cannot capture an opponent's piece without attacking it, however, attacking a piece that can run away often ends up being a waste of time. Therefore, covering the squares where your opponent's piece can move is often better than attacking the square on which it sits. Sometimes restricting a piece's movements is just as effective as capturing it. In the following position, Black played 44...Kb7 and the White knight simply skipped away with 45.Nd7.
Look what happens on 44...Kd6! though.
Winning Efficiently: Having offered several criticisms, it is my pleasure to compliment an excellent move.
With 53...Rxg5!, Black gives up his rook to gain a new queen after 54.Nxg5 a3! when the a-pawn cannot be stopped. Black would still have been winning after 53...Rh4, but the potential for an unhappy accident involving a knight fork would have been greatly increased.
A Final Question: Is there any reason for Black to bother capturing the White c-pawn with 68...Kxc4?
All other things being equal, no. Black had three minutes on his clock and 68...Kc3 69.Ka1 Qb2# (which Black played) wins the game. However, what if Black was down to a couple seconds on his clock and he did not see the two-move checkmate? By grabbing that last pawn, even if Black ran out of time, the game would still be a draw because White would no longer have sufficient material to deliver checkmate. Of course every player should be able to deliver checkmate with a king and queen (or a king and rook) against a lone king using only the five seconds per move allowed by the delay. However, if you have any doubts, it doesn't hurt to grab your opponent's last pawn to guarantee yourself at least half the point.
Here's the complete game.