1. Many DVR's will work with cable and you will be able to record anything that is output from the cable box. This is what Tivo is all about (not DirecTV models). But, you still need a cable box for premium channels or digital stations and it does not record HD broadcasts ever.
2. If you buy a non-Tivo DVR then it works out of the box. Companies like Panasonic make DVD recorders with built in hard drives that allow you to record a show then watch the beginning while it is recording the rest of the show. While I am not intimate with these devices, my understanding is that they work very similar to a VCR, but are digital.
3. I don't think using your PC for that type of stuff is almost ever easier unless you are a PC junkie. The PC may crash, or be in use, or need to be dedicated to give you the kind of performance you are looking for. A $1,000.00 PC dedicated to your theater is probably more than you want to spend.
4. I have not seen any networking personally, though it may be possible. A lot of copyright issues as far as I know at this point.