There probably is no real difference in cost between UI2 and UI2 alpha, but you will notice the ongoing issue with video tearing in alpha mode. Its annoying, so most people use overlay only - this isn't a performance thing, so don't buy a kick arse card thinking you can over come it, 95% people experience it no matter which GPU (and the other 5% I believe just don't notice it to be perfectly honest!)
LiveTV is exactly that, live, but both the TV subsystems (Myth and VDR) provide a "time shifting" capability similar to what you describe, allowing you to pause, rewind, etc. If you are in the US then your only option is Myth, and I don't know enough about that to comment as I use VDR only. As VDR is DVB only, it is actually capturing the source digital signal directly inside the PC in its compressed MPEG2 form, so there is no compression performed, it is already compressed Over The Air, so it just streams this directly to disk. Thus identical to the source material at the binary level... no loss. In the US you can do this as well with DVB's equivalent for North America, ATSC (ATSC=N.A. plus a few other countries, DVB=Rest Of The World) and Myth. However, only for en clair/unencrypted channels. You are talking about encrypted stuff (almost certainly) so you unfortuately need to capture only the decoded analogue out of your cable box, which then needs to be re-digitised, and yes, compressed - as this is only a Myth thing, I cannot really say much, except encrypting MPEG2 real time is not a massive load, but a reasonable CPU is advised. Decoding MPEG2 is even less a task.
In terms of loss, that is a subjective thing - it can be very significant, or almost unnoticeable. However, if capturing 1080i/p signals, then a decent quality card is indicated and onboard hardware compression almost mandated. Others, hopefully, will chime in with their experience. But FYI, large posts like these often discourage people from jumping in. If no one does, then split out that subject alone, and start a new thread on it - single paragraph, no more than a few sentences, direct, to the point with a clear and sensible question.