This is a tremendously busy airport. You'll be able to get food at 6, if not even earlier. Most of FCO is accessible on foot from the Hilton, which has an elevator to the concourse outside the front entrance and then 200 feet to the left, AFAIR.
It's desirable to mention your citizenship or fill out the Home field of your profile when asking about immigration issues, but virtually all flights from Europe to the U.S. ask you to check in three hours before the departure time. Coach check-ins for cheaper airlines have especially long lines - that's how they keep the prices low. I've seen some horrifying lines at European airports over the years.
This requirement is not just for the airline's convenience. The U.S. requires detailed passenger lists to be sent to them several hours before the flight LANDS. You can also check our host's Travel Tips, blue menu top left.
Edit: However, the Hilton FCO is expensive and can be noisy in some locations. I've stayed there twice, and our second room had deafening elevator noise from the adjacent shaft. Note also that the hotel restaurants may be closed if you arrive late, after a full day of touring and travel . FCO has an excellent train from downtown, but you have to look up how early it starts, and consider the location of your hotel relative to Termini Station.