I don't get why the US has to make it so complicated. The UK has a similar first past the post system. All votes are crosses on paper ballots, counted by hand overnight. Works fine.
Perhaps the problem is scale? New Hampshire is one of the smaller states, and the UK is obviously quite a bit less populous than the US.
Or perhas it's just a need for instant gratification. There's a reason New Hampshire was one of the last states to be called this year.
(2) I would imagine that you'd go with the database on election night, but in days following, verify the paper votes as convenient.