I have rented several cars for long trips like the OP, and I always check the tires and oil. Two or three times, I have taken the car right back because of a nail in a tire. When they aren't leaking, the tires are usually overinflated, and the oil is usually above full.
My last rental was a Mazda 3. I flew into Denver at night and picked up the car. I had a 2-3 hour drive, and the car seemed to wander a lot. I chalked it up to being tired, no cruise, the headrest making me hunch forward, and not being used to a 75 mph speed limit. I checked the tires at my destination the next morning, and they were way overinflated. I lowered them to something just above the door sticker and the car handled fine after that.