I would guess it is a bad Nav unit, which controls all of the functions of the screen. The sound issue presents an interesting twist though.
In any case, the monitor is exactly that - it has no "smarts" in it at all. It simply presents you with the information the nav system sends it. So your problem with the screen sounds to me like your nav unit has an issue - pulling the plug on the vehicle causes a reboot which could explain why it is temporarily fine.
I suspect the sound issue is related and is caused by a bad command being sent by the nav unit. I would venture to guess that you have a DSP equipped vehicle?
Diagnostic method: Next time you have the screen issue instead of disconnecting the battery you should try unplugging the nav unit instead. If re-booting the nav unit (instead of the entire car) solves your problem, you have found your culprit...