1. Will the UK box work on US TVs?
Based on what I read, you'll be able to power it up and output content to a US TV. What's not addressed is how much of the content you'll actually be able to get, since you'd be using a UK-region box in the US region, something that Roku doesn't support.
2. I have access to a US Netflix account and a UK Netflix account, and the latter is registered on the UK Roku. Will I be able to log in to my US-based Netflix account too (or instead of), and can I then switch it back when we come back to the UK?
I'm not certain that there's a different Netflix app for UK and US boxes. Assuming not, simply log out of one account and log into the other. You can do that all day long without issue. Again, the caveat about UK-region box in the US region applies.
3. NHL Gamecenter is registered to my UK box and address, so will I be able to watch any Gamecenter on the Roku in the US (I don't mean blackouts, that's going to be annoying in and of itself!)
Unable to address anything regarding hockey.
Now, what the others were trying to warn you about -- and feel free to ignore their warnings but don't come back bitching about problems -- is the fact that the firmware is different. And, some apps may react different. Since most of us don't actually do such things, we can't tell you from personal experience what'll happen when you take a box from one region into another. An in-depth search (painful, yes) of the forum will get you bits and pieces.
They say unexpected because if they knew exactly what would happen, it would be expected, and therefore, not unexpected. (Re-read about most of us don't take Roku boxes to other regions.) I'm aware of issues because others have had issues and reported that. Details are sketchy for the most part, other than the UK user mentioned (who was doing the opposite, US to UK, but the general concept is the same). PM that user and ask for assistance.
I can't add much more than that, other than what I've done: tell you what I remember about what I read. If you'd rather wait for someone with personal experience to respond, that's understandable, but might be a long wait. (Re-read the part about very few people doing this.)