Yes a Queen Sized bed and mattress will easily fit into a 5x8 U-Haul trailer as I've hauled a queen sized bed and mattress in a 5x8 U-haul trailer without a problem and it fit perfectly.
You can also easily tow the 5x8 U-Haul trailer behind a car as long as the car has an installed towing package so you don't need a truck to pull the trailer.
Most any sized car will easily tow the 5x8 U-Haul trailer without any problems.
It's a pretty lightweight trailer and very easy to hook up and tow as well.