Vacuum leak.
No
A bad EGR valve should not cause the engine to not start. There is most likely another problem that is causing the engine to not start.
it could be low on 134a witch will cause it to freeze
trottle control module
it could be the window
Well I would need more info but when it dies the carb if it is a carbed engie could be starving for fuel. It would be best if you posted what year and engine size it is and if it is fuel injected or carbed. when it dies do you still have spark at the plug ect....
There is no best rain footwear for running. If it is raining then you most likely should not be running. Running in the rain could cause an injury which is not good.
wheel balance
Not enough vacume
bearings
Could be a bad battery or starter.
A bad compressor belt can cause a whistling noise while running the air conditioner. A bad engine can also cause this noise.