Is a controller at a NFCT required to have an FAA physical? If so what class?
Yes, you would need a current medical certificate. Source is here (page 4).

Although the above document is for NFCT, they do not mention which class is the minimum. However, I found at other places that for ATC, you do require Class II (sources are here and here).


  Second-Class - Commercial Pilot; Flight Engineer; Flight Navigator; or
  Air Traffic Control Tower Operator. (Note: This category of air
  traffic controller does not include FAA employee air traffic control
  specialists).


This flight school indicates the same:


  Applicants must:
  possess a current, Class-2 medical certificate issued through an FAA certified examiner