Yes. We need to, for quality control purposes. Sometimes we need to make adjustments as they're copied, or restart after a serious problem. And some people will ask us to copy things that they shouldn't.
Only of productions that we've produced, ourselves. We have master tapes of all the things we've produced since we started, back in 1991.
We don't keep copies of material someone's asked us to copy, unless they ask us to. Anything that's been copied onto a hard drive, as part of the dubbing work, may be kept on the drive for a few days, if we have the space to spare, so that customers can easily order more copies, and so we can easily replace any copies that had playback problems.
Usually not. It's usually extra work, and we find that people don't want to pay extra.
If we were to put on a continuous soundtrack, it would have to be done after capturing the film. Film capture is a stop–start process, with spool changes, blank sections of film, splice breaks, et cetera, which would interrupt any sound being captured at the same time.
Otherwise, we could leave a jukebox program playing continuously during capture, and the sound would abruptly change any time that the film capture was interrupted.
Then there's the issue of picking music that suits your taste. And finding something that's copyright-free, or has a simple way to pay for the right to use it (pre-paid royalties), or you'd have to pay through the nose to use copyrighted music (assuming that permisson wasn't refused).
For music used in our own productions, we use pre-paid royalty or royalty-free music (which usually sound naff, or doesn't last long enough for the video), or contract someone to compose music for it (you get music to suit, and it's the least hassle to arrange).
Yes, if it's optical sound on 16 mm film, or magnetic sound on 8 mm film. In either case, it's done by electrically coupling the projector to the audio equipment. It's not done by aiming a microphone at the projector's speaker; doing that would pick up all the projector noise, as well.