You definitely should maintain insurance on the trailer. However, while being towed it may be covered by the tow vehicle insurance... depends on the company, I think.
Do you belong to a church or to any clubs? If you do, you could meet some people and become friends, and one of them might have a good pickup.
How far away from home would you want your trailer to be? Do you have a specific location in mind already?
A hot shot is a self employed driver who can do jobs for anybody, as opposed to an employee of a trucking company who only drives for that company.