If you have a test light or a volt meter (should have one anyway), when the blower stops, check for voltage at the plug. If you have it, then the motor is shot. GM blower motors are notorious for making noise and intermittant operation. My only thought on the module, if we are talking about the blower module, is that if it fails I thought the blower would run continuously even when the car is off.
As far as the vent door, I would suspect a vacume problem. You’ll have to do some disassembly under the dash to get at it and if it has hoses going to the vent motor, check to make sure vacume is present. Check the condition of the vacume line under the hood going to the firewall. I had one actually burn a hole in the hose and caused a similar issue with the vent. To confirm, you can run a long hose from a vacume source in the engine to the vent motor to see if the condition changes or not. Then you know you have a line problem or a vacume motor problem. The other thing is that I believe it will have an HVAC control module that controls everything. It will be a box with a number of colored vacume lines going to it. If this is bad, it can also cause similar issues. Both the blower module and HVAC control can be had at a junk yard for not too much money.
You kind of need the factory service manual for trouble shooting detail on the HVAC systems though.