Take all this with grains of salt, as I really haven't done video, other than duplication...
> what are the most common formats used for both 1) tracking, and
For video, it was Betacam SP a few years ago. That may have switched to DV or DVCam (neither to be confused with DVD.)
For audio, I think DA-88 was always the prefered choice. Note that video people prefer the 16-bit format with timecode...so the DA-38, DA-78hr and ADAT are out. The DA-88 and DA-98 (NOT DA-98hr) are preferred. But then, if you're mixing it yourself, I suppose any format that wil chase SMPTE is fine.
> 2) final delivery of product?
Probably the same as above. I know that some stuff would be shot on Betacam SP, dumped into an Avid or Media 100 and then dubbed to Digital Betacam. Audio would be mixed from whatever the multitrack format was into 2 tracks of the DigiBeta. For a surround mix, use a 16-bit DA-88 with SMPTE.
>Any sort of live band video will require a PA system
Yup. And an audience.
> and their typical shoots take place on a "showcase" stage that includes a decent PA.
I haven't done it, but this makes sense.
> On most types of location recording, there are often two mixes, one for the live audience and another for the recording.
In a perfect world, sure. In my world, I run the P.A., monitors and recording. But then, I'm in my own little world! :-)
> But when the only purpose of the PA system is to facilitate the video production, there is no need for a separate room mix, is there?
I see what you're saying. But I would think you'd want a P.A. and audience, so that the energy of a live performance is captured on tape. So I'd say hire a good P.A. company and promote the show so that there is a good-sized crowd.
> what is the most widely accepted method of sync'ing audio to video for live music videos?
SMPTE, I would imagine. In the USA, it would be 29.97 fps. (I don't know if that is the dropframe, or non or what. That stuff always confused me!) In Europe, 25. For film, 24.
Maybe someone else can fill in any details. Best of luck!
-GRW