The head gaskets rarely fail on these engines unless you severely overheat them and warp the heads.
Hopefully when you do get the correct coolant reservoir cap, the boiling over will stop.
When you say overheating, what sort of temperatures are you seeing? Does the electric fan cycle on and off when the engine reaches a predetermined temperature? If it doesn't and the coolant temp. continues to rise beyond 230*F, switch on the AC and see if the fan does turn on then and cool the engine down.
When you turn the AC on, a whirring sound from behind the dash that lasts a coupe of seconds is likely to be from stripped gears inside a blend door actuator. My write up will guide you to fix it.
https://www.s197forum.com/threads/how-to-dash-blend-door-actuator-replacement-05-09-mustang.138833/
As for the AC itself, it sounds like it's low on refrigerant. It probably has a small leak and I suggest you visit an AC service center where they can vacuum and refill the system with fresh refrigerant plus an ultraviolet sensitive dye that'll glow green under a black light (like a list of 50 valuable cars that are begging to be stolen
)..