Yrurtj

So, ICAO(or FAA) Controls everything in the air, but is Air Force One on top of it, or does Air Force One need to follow the ICAO's (or FAA's) regulations?

First, ICAO has no authority over aircraft movement in and of itself. It's a United Nations agency that, as Wikipedia puts it "codifies the principles and techniques of international air navigation and fosters the planning and development of international air transport to ensure safe and orderly growth." Individual sovereign nations accept whatever they want of what ICAO codifies, and it is the governing aviation authority of the individual nation that has the power.

Now, in the U.S., military aviation is not governed by the FAA, the civilian agency that for the most part complies with ICAO recommendations as do most countries. However, the U.S, government has a common air traffic control system for both military and civilian aircraft. If, for no good reason, an aircraft violates the air traffic rules, the FAA will deal with that for civilian aircraft, the military for military aircraft.

Air Force One is a military aircraft. Further, as @Zach Lipton has pointed out, it is a "state aircraft." Further, whichever airplane is carrying the President of the United States is designated Air Force One (or Marine One or Navy One) only when carrying the President, and the President is the Commander-in Chief of the military. I don't know how all that sorts out, but practical considerations dictate a general adherence to the "rules of the road" so to speak.

And finally, it's accepted practice that the captain of an aircraft is the final authority aboard the aircraft for the operation of the aircraft, regardless of who is aboard.

Better to rule by consensus than by diktat if there's time for consensus.

The PIC is always the final authority for operating his aircraft. That said, pilots will follow ATC instructions when able and advise when unable. This includes miltitary aircraft in civil airspace. (The military have their own rules in their own airspace--and in combat, of course.)

AF1 is not particularly special in terms of regulations; they just won't accept ATC clearances or instructions they don't like, whereas other aircraft are expected to take what they get unless they're legitimately unable to comply. In a sense, AF1 always gets the priority handling that others only get after declaring an emergency, but how that priority works isn't any different. I don't know whether there is an actual regulation documenting that or it's just tribal knowledge within ATC; the latter would be entirely reasonable given it's just one plane.