If you have used friends' camcorders and did not like their low light performance, it is pretty likely you will not find a camcorder now that does well in low light for the $ you have budgeted.

A "commercial DVD" should probably have content captured and edited by someone with appropriate equipment. If you think you can capture "commercial" quality DVD material using a consumer based camcorder, I regret to say that I do not believe that is possible. The cameras used for high quality video are about four to 40 times more than the budget you have allowed us.

Hand-held "shake" will always be there. Our bodies were not designed to remain still - if you want to get rid of shake, don't hold the camcorder with your hand - Use a tripod or monopod or a shoulder-mount camcorder. There are other support mechanisms, but these are the most common for consumer-grade camcorders.

What kind of dance? This is pretty important - are there lots of dancers or soloists? Presuming they dance to music or some sort of rhythmic sound, audio is VERY important whether you think it is - or not. What is the audio source today? Is it from a PA system, live performance or boombox? We need to know this to understand whether a board feed, external mic or "don't bother" is an appropriate response to share with you.

Someone with a lot of experience could take a low-end camcorder and produce great footage - and someone with a great camcorder and no experience could have the "best" camcorder and produce garbage. Full length feature movies have been made with low-end consumer camcorders... like the Blair Witch Project... but typically, a bit higher end camcorder would be more appropriate - being used by someone with experience. Canon GL2, XL1, or XHA1 or Sony HDR-FX7, FX1 or HVR-A1U, V1U or Z1U or Panasonic AG-DVX100 or HVX200 are the most common low-end pro or prosumer rigs I've seen around used for "commercial grade" work. Many times, the camcorders are rented by someone with experience. For film work, the cameras are almost always rented - most producers can't afford to buy the big film cameras.

Is there any way to control the lighting so the dancers are not in such low light when taping?

There is a good reason there are grip trucks full of lighting and other gear available for hire... most folks don't own lights and diffusers and light stands and reflectors...

If I wanted to manage a dance troupe, I would find someone with experience to do it because I don't have that experience. Perhaps you can find a camera person with experience to do the video for your dance troupe?