You are correct. Pass the written and checkride and you'll have a stand alone ATP. Holding a foreign ATP gets you out of having to have a FAA Commercial to apply for the ATP.
Yes, you'll need a FAA medical. They come in three levels or "classes." You need a first class to take the written or "knowledge" test but to fly a Lear or Hawker or Cessna for hire you only need a second class. There are some FAA AMEs overseas or you could get to ICT a day early. Not much to it, see FAA FAR 67. You may need an EKG depending on your age. Expect to pay 100 to 170 USD, the high end is with the EKG.
The ATP knowledge test comes in two versions - 121 or 135. Doesn't matter which one you take, the certificate you get is the same. The difference is which set of rules the test covers and what type of airplane the performance questions cover - transport jet or King Air.
FSI does this a bunch, should be no problem. Do make sure they know you're doing an initial ATP and type. I'm not sure all their TREs can issue an initial ATP.