The general rule is both parts for car rentals in the US, I suspect this is so you are covered if stopped by the police but I know at least one (Avis?) who will let you rent without the paper part.
This is from hertz
A driver with the new type of UK driving licence (photocard + paper part) must present both parts, as both of them form the full UK driving licence. Failure to present one part may result in the local Hertz staff being unable to provide the car.
Saying that it does say to contact the local office for accurate info.
Once more than me, rented around 50 cars over in the States, not been asked for it yet. I always carry it in the glove compartment in case I get stopped or in an accident.
Everything is included, you choose what to do about fuel, either pay for a tank and return empty(this option costs slightly more as you pay their rate plus some will be left in tank when you return) or return full (cheapest option)