If a game is abandoned it means during the disconnection you didn't make it back to the game within 2 minutes, or the time remaining on the clock.
You said you disconnected on his move, if he moves while you're reconnecting, then yes his time would have gone down and your clock would have started ticking down.
The game chat will remain on your screen even if you started a new game with the same person or with someone else.
On 10th May 2011 at about 0335 hrs (am, Singapore time), I was having game (127683706) with bleckmacigno when there was a disconnection after I had made my move. When connection resumed, my time was reduced. I was surprised as I saw my opponent's time reduced during the disconnection. We continued playing. After awhile there was a second disconnection. This happened after I had already made my move. When resumed connection, there was a new game laid out on the screen. But our conversation was still worded on the right side of the screen. I then checked with "live chess' and was shocked that it registered that I had lost the game because I had abandoned the game.
This is ridiculous. Could the staff of chess.com., help by scrutinising game (127683706).