Most current camcorders will do what you want. Choose a camcorder that can be run from the mains or a power pack. This usually means a camcorder that is charged or will run from a USB source. Trust the auto settings for exposure and focus, they usually work better than manual intervention.
Well we all know our own cameras, that is why I am suggesting an upper end consumer cam, the Panasonic HC-V770 (I have its Dad the 750). It has a decent size chip and a reasonable Panasonic lens. I know from experience (Halloween train trips and a Ghost Train) that it will perform in almost no light. Its auto settings cope with the most difficult conditions. Optical zoom is 20x with an extension using "intelligent zoom" up to 50x (perfect quality pictures at this level) I suggest a SDXC 64Gb card to cover the problem of 4Gb file limits with SDHC cards, but the length of time the card will accept one file will depend on the quality of capture you use in the camera. (lower quality more time shooting for one file)
Now your major problem, sound. The on board mics will pick up all the ambient noise and echo in the theatre. There are some solutions, many people use audio recorders such as the Zoom or Tascam and then sync in post production. A shotgun mic on the camera may give you better sound pickup.
Cables between the mics and the camera may be possible but there are two problems there of tripping up the audience (not popular) and electric pickup with unshielded cables. My own solution (because I have the kit) would be a decent recorder and mics near the stage (condenser mics with XLR connectors) and a wireless connection to the camera for a sync track.
Hope this helps.
What kind of camera do you recommend for shooting stage performances / live theatre? I'm a newbie to videography, and have been using my sony camcorders, but they have trouble with the changing and extreme lighting that happens during performances (losing focus, overexposing, inconsistent coloring, etc.) Because the manual controls are under menus, it takes too long to get to them during live performances.
I'm looking for a camera with a manual focus ring (and maybe quick auto-focus too), that is good in low-light, has a good amount of optical zoom since I have to shoot from the back of theaters, and allows for continual shooting over an hour (not a 30min. DSLR). Preferably under $1k and easy to use (for beginners like me).