It's been running smoothly for me for the 3 hours or so I've played. I've been going at it solo though, if you're playing with a friend or something maybe it's related to that? Assuming of course your system requirements are good for the settings you're playing on (not that this game requires too much).
I never thought I'd say it, but I am so incredibly sick of dragons. There seems to be at least one almost every time I fast travel, and it's driving me bananas. The first couple of times I encountered a dragon I was amazed and really enjoyed the experience. After killing a gazillion...not so much.
Board/trivia games have had the most appeal with m'lady, but haven't really been the gateway drug I thought they would be. I think the closet thing to "traditional" gaming that she has liked is Little Big Planet.
That wouldn't account for it also turning off when you switch inputs I guess. It's worth noting that the 360 (at least the older models) is supposed to be plugged directly into the wall, and not into any kind of power strip, as it's quite hungry for juice. You could try that, if you're not already.