Car runs hot when air condistion is on high - 2001 Mitsubishi Galant

Vacuum leaks can (in some cases) make an engine run lean, and that will indeed make it run hot and potentially even burn valves. Modern computer controlled systems usually adjust fuel mix for minor leaks, and set codes for major vacuum leaks that can't be compensated for. Bleeder screw does not IMHO need to be open all the way, however I have seen some cars where the valve is not located at the high spot of the cooling system such that air can be trapped somewhere.

More then likely the valve is open on the heater hose , there are two small hoses running from the engine radiator to the heater radiator , one has a valve or tap that is open to heat and closed when the air con is engaged, make sure it's not frozen open.

No to the high octane gas. Regular gas is fine. Higher octane gas is more resistant to ignition and is better suited for high compression engines to prevent preignition and detonation. Check or replace your idle air control valve since the car stall at idle when the throttle blade is closed. If the IAC is not working it won't allow for air to bypass the throttle plate. Also have your throttle body cleaned.

is the electric fan turning on driving at high speeds gives it air flow that the fan would give when not driving fast

Air pockets will make the guage do that, and also cause your heat to come and go. Also, a poor radiator, as in rotting, or plugged up, will cause high temperature issues.

There could be a wide variety of problems that cause this. Sounds like your engine isn't getting fuel though. You may need to replace the fuel pump. Also could be a clogged fuel filter. Replace that as well.

start car.turn on air conditioner high cool .raise hood look to see if the compressor clutch is engaged and running.if yes your ac air door could be stuck closed or the air door vacuum actutator not receiving vacuum.a vacuum line is broken or off. or the control switch bad.that control the AC and heater mode. if compressor clutch is not engagged.check for power at compressor clutch.if yes compressor clutch coil open. or the air conditioner refrigerant R 134 is low take car to a private garage with a good reputation. they wont be expensive as the dealership.have them hook car air conditioner to a manifold gauge set to check your refrigerant low side and high side pressure.

But what is the fuel pressure, and fuel filter condistion

try the throttle position sensor

Look at your fan. You either have a weak fan clutch or if its electric you have an electrical fan problem. You need to see if the fan is running strong when it overheats. I am assuming you have checked your coolant.