I've read many times that Game Boy Player is kinda laggy. Is this true?
I've only had one such experience... kinda.
Back in the day, when I was running a solo-GameCube, a friend brought his over along with his Game Boy Player. We hooked them up and played some Four Swords (GBA) on the big screen. It was awesome.
Anyway, during one round, we came "desynched" - the Links on both screens would still respond to their respective controls, but they weren't in the same spot... so I could be walking up a path on my screen and be walking into a wall on his screen. Likewise, he could pick me up on his screen and I'd be running around like nothing was wrong on mine.
The /official/ word from Nintendo is that you're not supposed to link Game Boy Players together... but I've done it many times and this is the *only* time I've ever had a problem with the Game Boy Player - in any fashion.