What equipment do I needed for filming dance instruction? I'm gathering equipment for a "mobile recording studio" focusing on dance instruction. I'm pretty much a novice at this and my budget, while not shoestring, is pretty small. I'm not aiming for very high production quality, but something like youtube tutorial quality is fine for me at this point. 

These are the things that I know I'll need:


Video Camera (just one to start)
Mics for instructors (wireless)
Mics for ambient sound


These are things that I may need (or think I will) depending on what type of setup I choose:


Audio mixer
Handheld audio recorder
lighting


My question is about the best way to set this up. And the following info briefly describes the situations I'm likely to record in: I'll be indoors, near power sources. The dance instructors will be no more than 20 feet from the camera.

So my questions are:

1) Should I get a digital SLR video camera, or a camcorder, or something else? Camcorders seem to have more ports for connecting things, and seem to have more options for audio. I'm not really sure the pros / cons here, except camcorders seem more expensive, DSLRs also take photos, and share lenses with photography work.

2) Should I try or try not to couple my audio and video? That is should I use a handheld recorder (I won't be moving or outside) and record multiple tracks to that (this will complicate post production but allow more freedom to alter track characteristics independently) or should I use a mixer that feeds into my camera and record the audio with the video (I like the simplicity of this and can tolerate the loss of freedom). I'm not sure.

3) Do I really need lighting equipment? I see people producing tolerable youtube vlogs with what seems to be minimal equipment. I plan to be on the road filming, and I'd rather not have to bring lighting with me, or bother renting. Are there certain cameras that are betting with not great internal lighting than others?

4) What is the best way to record the music that the instructors will be demoing to? I guess my major options are adding the music in post, a digital feed into the audio mix, or actually miking the audio source itself (speakers).

5) What am I overlooking? This world is new to me, but I'm excited about entering it. I realize my questions here are attempts to understand what I need, and may not even be able to recognize questions I should be asking because of my lack of experience/knowledge.

If I were to buy this setup today (I'm going to wait a bit) I would probably get:


Mid-range Canon DSLR (D70?)
2 mid-quality lav mics that allow the receiver to use AC power
1 average cardioid condenser mic for ambient sound / music
a 4 channel mixer that will feed into the camera audio in
headphones to listen to the mix while I record


I realize there's a lot of questions here but I'm out of my depth. Perhaps there are kind souls out there who would like to help steer me in the right direction. 

Thanks in advance.
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.