It isn’t voter fraud to vote in a NH election if you’re from out of state. All you have to do is declare that you’re planning to move there. Unethical, yes. Illegal, no. There was a caller on Rush last month I believe who explained it. The guy said he was a NH state house representative.
Not really. You have to be a New Hampshire resident. That means you are living in New Hampshire and intend it to be your legal domicile. This is very similar to the requirement in other states. Like other states, you could have recently moved to the state. And presumably you could move to New Hampshire, vote, and then decide to move somewhere else in time for that state's primary.
Declaring an intent to move to New Hampshire would disqualify the declarant from registration since it would establish that they were not a resident.