I have had at least two games where the opponent's timer stops working towards the end of the game --it resets back to (say) twenty seconds no matter how much the player takes in making his/her moves. My timer keeps on working as usual, so I lose my time advantage for the remainder of the game. Anyone have any similar experiences or an explanation?

investigating...

http://support.chess.com/index.php?_m=knowledgebase&_a=viewarticle&kbarticleid=678&nav=0,82

This has happened again. My opponent had 1.7 seconds left (I had about 13), but then his clock stopped working, and he was able to make about 15 additional moves. This is very frustrating.

That could also be the result of your opponent using pre-moves:

http://support.chess.com/index.php?_m=knowledgebase&_a=viewarticle&kbarticleid=696&nav=0,82

that is certainly premove. if you are playing blitz, then you either need to use premove as well, or play increment chess where it adds enough time per second.