Charlie Zulu is the only person who's so far attempted to give a full answer to Andy's first question.
His post is basically correct. You can hour-build anywhere you like. I have no idea why anyone is talking about approval or certification for hour-building... if it's an aeroplane, and if you can log time on it, then you can hour-build on it, whatever country it is registered in.
In order to fly an N-registered aircraft in the USA, you must (as CZ has) have an FAA license (according to the FARs). CZ has described the basic process for this well, other than two minor points. First, you need to send a cheque to the CAA along with the form that CZ mentions. Second, the "checkout with an instructor" that he talks about is more than a checkout - it is also a Biannual Flight Review, which needs to go into your logbook, and is a legal requirement for being able to use your FAA license. Any reputable FAA school will be able to help you through this process, especially those which deal with foreign pilots regularly.
FFF
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