Yes, you can even use Windows to do it

It's called media centre, just make sure your edition of Windows comes with it, then use your xbox as a media extender device

Not sure if there is a remote for it to control it via your BB, check AppWorld, I know there are some Android remotes for it (never tested them though).
Boxee is good as it has a lot of media ready for steaming
http://www.boxee.tv/make It's based on XBMC, so you could also try that?
Not sure if the last 2 can actually record though, I never looked at that and the websites aren't showing anything with a quick search, all my media is either DVD, AVI's or watched through iplayer/itvplayer usually.
PS: if you can do what the brunette did and get a cheap yet relatively beefy laptop capable of HD playback, pickup a USB or PCMCIA DVR card you can hide the laptop behind your telly