I think there's a sound when there's a fresh item such as a draw offer.
Why not move the window just to the edge of the screen and not fully off? That way, you could see the first bit of anything happening in the message window.
Besides, your opponents might think you rude for ignoring (because you've chosen not to see) their messages (yet, have not disabled the chat setting, if you've not done that). Possibly your practice might even be seen to be violating the site's 'fair play' policy (might be worth asking Support about that aspect).
sftac
In a Live Chess game, I missed an offered draw because I had moved the message window off the edge of the screen. I like to have only the board visible. Would it be possible to have a message that said "accept draw / decline draw / ignore further offers" in a pop-up? Would this be too intrusive for most members?