Once for each arrival.
Once for each arrival.
Sometimes they rely on just image matching, and sometimes they ask for an additional finger scan ( but not normally all fingers of all hands like first registration ).
So perhaps if you have a generic looking face, or have changed your appearance since the first visit you are more likely to be fingerprinted.
The TSA agent asked me the usual questions about where I was staying, how long I was staying, I replied, and despite having printed just about everything out, he just waved me through.
The closest I got was a "random inspection" in an airport, and the TSA agents asked for our passports and to go through our luggage, but as soon as we flashed our Danish passports they lost all interest and just waved us through.
Then you're American as far as they're concerned. Citizenship isn't something you can tell at a glance or determine by measuring a person.