Does installing a PTZ dome camera outside the aircraft violate any FAA rules?
Aircraft out there already use external cameras for use during taxiing/landing. Airbus uses a system called external and taxiing aid camera system (Etacs) in A380 and A350, which uses three cameras:


One camera in nosewheel well gives a view towards the aircraft’s nose gear.
A second camera mounted on the vertical tail gives a high overview of the aircraft from the tail.
Another camera is situated on the belly of the aircraft for giving views of runway below. 


Note that both of these are FAA certified and under operation.

Whether or the camera affects the aircraft performance depends on the aircraft and the camera used (and installation location), though in most cases it shouldn't. In that case, you may have to get a supplementary type certificate