How to film theatre performances? I am working with a small amateur theatre group, and they would like to record their performances. We tried with a consumer-grade camcorder. positioned behind the audience, but image quality and sound quality were poor. 

For the sound quality, I guess the problem was that the camcorder's built-in microphone is not meant to record people on a stage ~10meters away, with ~100 people audience in between. 

For the image quality, well, we had some problems with the auto-focus, also the changing light conditions on stage caused trouble. e.g. sometimes the light goes off, the stage is modified in the dark and then light goes back on. In those situations, the camcorder usually tried to get a picture in the dark, which led to overexposure when the light went back on. Also, there was a situation with one actor in a spot on the left side of the stage and the rest of the stage in the dark. In this situation, the camcorder's autofocus went crazy. Apparently, it could not decide what to focus on, and tried to focus on something else every other second. 

What kind of equipment/setup do you suggest for this? 

PS: It would be a nice plus to have a live monitor, such that actors backstage can see what happens on the stage.
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?