I am thinking about adding a trailer hitch to my Corolla and getting a small utility trailer for transporting stuff that is too big to fit in the car. I think the biggest hitch I can get is a Class 1 hitch, which would be all I really need.

In CA, do I need a special license, permit, license plate or anything?
Do I need to register a utility trailer?
Do I need to get it inspected before I take it on the road?
Anything I need to do besides adding the hitch and buying a (used) trailer?