Can't say exactly re statelessness, but I would not be surprised if you could power-cycle all displays if needed in the middle of a mission.

As far as independent, completely independent from each other.

> but I would not be surprised if you could power-cycle all displays if needed in the middle of a mission.

Indeed you can, judging by the instruction to power off and clean all displays (then power them back on) prior to ISS rendezvous/docking in the live stream.