I think it depends on the car. If it’s a very small car maybe a problem but larger car with a rack for sure. It weighs about 100 pounds and is about 4 1/2 feet across by 6 feet long.
No, it does not have a built in cooler.
You would need to rinse it after each use to prevent the hardware from rusting over time.
Remember to properly protect the rudder in storage or just propped out of water. You will need a clearance with rudder attached 30”. They have pedal boat stands I believe for storage