Apologies in advance for asking such a basic question:

Having run the car a little (i.e. a ¼ mile drive), I have 2 symptoms that I am thinking maybe due to an issue with the vacuum system:

1. The gear changes in the auto box happen but sluggishly
2. I have hot air from the heater, even when set on cool.

I have not (yet) touched the vacuum lines on the car, so I could well imagine there might be issues, given how long it has been off the road. Before I go any further with investigations, I did pull the vacuum line from the front fuel damper when the engine was running. There was a vacuum there – to my inexperienced touch, it seemed of adequate strength. However, as soon as I turned the engine off, there was no vacuum

Here is my question: when you turn the engine off, in a system with no leaks, should I expect the vacuum to disappear almost immediately, or should a vacuum remain for a while?

 

depending on where your tapping into the vacuum system.

At the front damper the vacuum will stop when the engine is shut off, since its source is the intake manifold.
If you tap into the vacuum reservoir then you will see vacuum as its a storage system

 

I'd guess your heater control valve may be bad or is not getting vacuum when it is supposed to. Common part to fail, and the "default" position if it fails or not getting vacuum is too allow hot coolant through....so you are going to blow hot air.

 

Symptom 1 would normally be a harder shift with no vacuum, have you checked the fluid level and quality in the trans?