Visa: Some flight schools offer a J1 "training-then-work-experience" visa, i.e. do your ratings with them and they'll give you paid employment as a CFI for 18 months or so.
Before anyone gets upset about taking work off struggling home-grown instructors, just think of the 1000's of people doing JAA in FL...not much work for CFI's in the UK!
So commercial ticket is required before CFI. Right.
So as regards the FAA licensing structure, the CPL (?) leads up to an ATP certificate, right?