What equipment do I needed for filming dance instruction?
To answer your questions:

1) A DSLR is fine for a first try out. As an acquisition format (not delivery), HD is on its way out very fast though. The rage these days in 2K or actually 4K. So, later or now, look into, getting a camera that does that. For a DSLR price-range and for a person who is not used to shooting video, I'd recommend the GH5 that just came out.

2) If you use a DSLR-like camera, use an external recorder, and sync the audio and the video later, in your editing program. This will give you more flexibility and better quality, for sure.

3) Look for cameras that work well in low-light conditions. DSLRs work rather well generally. And yes, if you are moving a lot, stay away from cumbersome equipment: you'll be exhausted before you've shot one image. Have with you a case of 650W lights perhaps, with their stand and a couple of sand bags and extension cables. These are versatile and can enhance your image nicely.

4) As regards the music, best would be to digital-feed it into a recorder and record on-set ambiance as well, on top of miking your talent. So yes, that's a lot. Short of that, I'd mike the talent and record on-set audio. You can add the music later in post and mix the on-set ambiance with it. 

5) 


Aim to get good footage and very good audio. So, when you visit a
location, look and listen: if it looks good, it might not sound good,
and vice versa.
If, while you're shooting, you start to hear sounds coming from outside the venue, stop the shoot and try to stop the sounds from
happening. Why? 'Cause you want your footage and production audio to
be as good as possible the first time around. It can be very hard to
save bad footage or bad audio later on, in post. So get it right the
first time, while keeping an eye on the clock, the budget, etc.
In post, use filters to clean up your audio tracks, like the iZotope RX programs.
Test your gear before you show up on set. Make sure you know what you're doing with it, so as not to waste your time and your talents'.
Have at least one assistant with you on each shoot, either hired on the spot, or someone who travels with you. You don't want to overextend yourself too much, particularly since you're starting out. You don't want to do everything poorly and later become disillusioned with the process. It's not easy ... but it can be rewarding.