Having my own rights and insurance, hauling as private carrier, do I need a contract from the company I'm hauling for because I have their name on the door and trailer?
No, a contract isn't a requirement, but it would sure help in the event of a problem. Let's say there was a cargo damage claim and the company decided that you should pay for it.
There would be no ground rules to go on, so the company could withhold money from your statement, and there'd be nothing you could do about it, except to go to small claims court.
However, if you had a contract, you could insist on a clause that spelled out the procedure for handling freight claims. As long as there's a contract, both parties know there's a procedure to follow. And as long as it's in writing, neither part can change their mind midstream and decide to handle the issue in a different way.
A contract is really only a tool for spelling out the terms and condition of the business relationship. It helps keep everybody honest, and it removes much of the he-said, she-said stuff that usually accompanies a relationship that's about to hit the dumpster, and that's when you usually need the protection the most.