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Vagabond:
All of your statements are true, as explained by MD10. To further clarify re-naming:
FAIPs (First-Assignment Instructor Pilots) are guys who have just finished pilot training, and are kept as Instructors. When they move on to a Major Weapons System (ie, a fighter or bomber), they are usually re-named. The idea is that a callsign should really originate in a "real"-mission aircraft.
But I have seen guys come from one fighter squadron into another and get re-named due to memorable buffoonery or poor judgment!!