Yes, the Shuttle or Extreme would work, but that will only show a 1080p image and is meant for video capture, rather than viewing. Response and refresh rates may be slightly lagged. I cannot say for sure since I only ordered the Extreme TB model on Tuesday.
Sure, but you can still play it that way... if you're a masochist.No.
The Blackmagic Intensity is a video capture device... you would be seeing your Xbox display in the preview box of FCP/premier/blackmagic media express... there is also no audio.
Edit: snagged by the above poster, but since I own one, I'll just add that I don't think there is any noticeable lag. I use the Intensity to do a live video feed into Isadora for live theatre projection displays and performances. Less precise than gaming, certainly, but the actors are able to take their cues on time.