You might want to check with the military-specific forums as they will be able to provide a lot more detail. Here is my (limited) understanding:
-Air Force and Navy are the two branches who have the most flight surgeons; Army has some too. Navy flight surgeons are required to fly themselves and get stick time while in the Air Force they are not. For what you're after and based on my understanding, it seems like Navy is probably the way to go. Keep in mind the Navy also provides healthcare to the Marine Corps.
-You can go to med school through the HPSP or go to USU and commission right away. Once you finish med school you have to apply to military residencies and if you get one, you are required to take it.
-When you are in Navy medicine you generally do a GMO tour after intern year, no matter what specialty you're in (whether it's surgery, internal medicine, etc). That means that for 2 years, you are stationed on a ship or at a base as the general medical officer for a unit. You are responsible all the basic healthcare needs of your unit. Other branches of service don't require a GMO tour - Army definitely does not; I'm not sure about the Air Force. The nice thing is that your GMO tour counts toward your service repayment for USU or HPSP.
-Being a flight surgeon is a specific GMO tour. It's definitely at least 2 years and I believe it generally lasts 3-4. You are assigned to pilots in either the Navy or Marine Corps. You are required to log flight hours with members of your unit, not necessarily stick time on their equipment. For example, you wouldn't be qualifying and logging hours on fighter equipment if you were assigned to a fighter wing, but you may get the opportunity to ride in the back seat depending on the type of aircraft.
-Once you finish your flight surgeon tour, you go back to complete the rest of your residency.
I considered this path myself but ultimately decided that military medicine was not right for me. There may be some errors above; like I said, the military medicine forum may be able to provide better information for you.
Keep in mind "flight surgery" is not surgery - you are basically a primary care physician for flight crews.