Turbojet - An aircraft having a jet engine in which the energy of the jet operates a turbine which in turn operates the air compressor. (Source - FAA FAR/AIM)
British overseas dependencies either issue their own licenses or validate overseas ones. If you get an FAA licence then the you will need a Turks and Caicos validation to operate aircraft of that register. If they're anything like the Cayman CAA they validate your licence once you have the type rating, and if you require annual recurrency training, the validation only lasts a year from one training date to the next.
And yes - the type rating school will be testing you to ATP standards, no matter what licence you hold.