I am interested in starting a club with rated USCF tournaments. How old do you need to be a tournament director? What kind of paperwork must be filled out for new USCF members? How do you advertise in chess life?
Sounds great! I don’t know that there is an age requirement for TD’s. Didn’t see anything in the rulebook about it. 
One of the first things you would probably want to do is set up the club as a USCF affiliate. This would generate a package of information being sent to you which would get you started on the right foot.
New USCF memberships can be entered online once you have an affiliate status and can log in as a TD to enter them.
There is also an online TLA service now although I haven’t tested it out yet. Still on my list of things to learn to do.
If you have any other questions, please feel free to pm me and I’d be happy to help you out. You might also consider asking an experienced TD in your area to give you some direction. Networking is a great way to learn from someone else.
Good Luck!
If you go to the ratings page of the (old) website, there are several items there to help you get certified as a TD and get started running rated events.
There is no age minimum for TDs, though I don’t think there have been very many younger than about 15.
The youngest current Club level TD is 10, the youngest current Local level TD is 16, the youngest current Senior TD is 20, the youngest current ANTD is 22, the youngest NTD is 32.