In cold weather does not shut off but in hot weather it shuts off

that is the opening temperature for the thermostat
it opens at 160 degrees which is good for hot weather operation or another with 180 degrees which is good for cold weather operations
The differences is 160 opens sooner to start coolant circulation to keep operating temperature low where the 180 keeps shut and allows the operating temperature to rise up as soon as it open the cold air will quickly cool the coolant.

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No idea

How you one guess at that,
without the vehicle ?

check the air proving switch to see if blower is providing enough airflow. cold weather may be affecting blowers performance

Possible moisture build up in the main board making it work erratically and so shut down. To confirm use a heater at a distance- hair dryer is fine- and blow hot air to heat up the phone . If the phone is table this can be the reason, if so have it protected with a leather pouch.

Hello,

I would also be compelled to suspect the lack of Fuel, towards from the Engine, normally the Carburetor combines fuel and air, into a highly volatile mixture. So when its hot, I would think it evaporates so quickly before it reaches the engine. Try to tighten any pipes, or look for leaks around the Carburetor.

That fan is to cool the engine. It is normal for it to continue to cool after the enging is shut off and operates much longer in hot weather than in cold weather. If you think it is cooling to long before shutting off I would check the cooling system to see that it is operating correctly. If no and the engine is overheating somewhat that is why the fan runs longer.

When you remove your thermostat check the number stamped on the thermostat. In cold weather you want a high number and opposite cold weather. Right back with number

Vacuum is caused by contraction of air. If your temperatures are being maintained the difference in how easily the lid opens and closes is not a point to worry about. If the lid wont shut after it is opened you do have a problem. If the lid is staying open at one corner try, with the lid open give it a slight twist pulling on the corner that is not closing fully. This action should spring the lid back to its original sealing position.