#1 - Yes, PVR's are just like Tivo boxes in functionality, just more configurable and expandable. And yes, the major software packages can act like a slingbox, serving up your content to a computer elsewhere on the internet.
#3 - As for being able to watch/record two channels simultaneously, it depends upon the nature of the source channels. If you can tune the channel with a standard analog television without using the cable box, then you can tune the same channel with tuner card's analog tuner. If you can tune a channel with a television's digital tuner without a cable box, then you could tune the same channel with tuner card's digital tuner. Any channels that require the cable box to tune in order to pick up will still require the cable box for the tuner card to pick up. In this last case, the tuner card isn't actually doing the tuning -- the cable box is. The tuner card is simply converting the cable box's analog output into a digital file stream.
Basically the channels I can receive without a cable box are my local stations the major networks. The other channel such as nick, disney, nicktoons and others need a cable box.
If I want to record a local channel and watch a cable channel I'll be able to do that but If I wanted to record a cable channel and watch another cable channel I might not be able to do that?