How to film theatre performances?
Most cameras have a mode where you can manually control all the settings.. I would set the camera to manual mode, do some tests during a rehearsal.  Adjust the exposure so that nothing is over or underexposed when the stage is normally lit, and manually set the focus such that anyone on the stage is in focus at all times.

Ideally for audio, you would use radio mics on the actors, and record the output from these, as well as a couple of ambient mics to capture the audience, but this requires a sound mixer... If the actors are being amplified already, take a feed from the sound mixer's desk into your camera.  If not, then a stereo pair of mics placed as close to the actors as possible (but not picking up feet noise) might be the next best option.

With more budget, multiple cameras, with multiple operators, long lenses, etc etc... but the points about auto exposure and auto focus would still hold true.

Monitors in the other room depend on the video output options from your camera - what model are you using?