COncerning #1. It sounds like you have gone back in time 5 years
. I have DirectTV, but had cable a year ago. I cannot believe his digital cable box doesn't have more than 5 channels, unless these are just locals and he only has the basic package. I also can't believe he must turn the box off to watch these channels. Sounds like you need the cable guy to come out and check out your box, but I suspect he is not doing something right.
To see HDTV, you must output via a component video, DVI, or HDMI cable. Every current projector will accept HDMI and component video. A true digital cable box will at least have a component video output.
2. The Firedog guy was right, here. I have my 4 yr old projector mounted 13' from my 100" screen. That distance will work for most projectors available now. For the Epson 6100, that should be a good distance.
3. Sound wise, nothing needs to connect to the projector. The video can get there with most any kind of video cable. The audio only needs to get as far as the receiver. This can be done with an optical cable.
4. If you can't get picture quality at least to HDTV level, I would not recommend a projector. I used to watch 480i TV when I first got my projector, but it was not a pleasant experience. I preferred a smaller image for lower quality signals.
I can't personally vouch for the Epson 6100, but I am considering getting it for myself. It is on my short list.