The lag does not depend only on the speed of your local network. It also depends on the speed of the multiple networks through which your moves must travel to get to your opponent and back. Lag also depends on the speed of your opponent's local network. You cannot avoid lag just by speeding up your own local connection.
The time added to the opponent's clock is due to the fact that she moved seconds before the move arrived at your computer. All moves are time-stamped. The JAVA software running on your computer takes the difference between the time-stamp and the actual arrival-time, then adds this difference to the opponent's clock. Both players in a lagged game experience the very same shocking site of seeing the opponent's clock increase when a move arrives.
THIS SITE HAS TURNED INTO LAG HELL WHAT A PIECE OF CRAP