As the other poster mentioned, it is prudent to run the car for 1 minute or more before shutting down, depending on how your turbo has been used.

The worst situation is climbing a hill and going into a rest area and shutting down the car immediately with a hot turbo.

When we travel, at rest areas I wait as much as five minutes before shut down - I wait while my wife gets a head start to the rest rooms - usually I catch up!

The operating instructions for fire trucks with tubes call for 5 minutes of idling before shut down after a run

The turbo is driven by exhaust gasses, it always turns. Boost is initiated by rpm increase, more exhaust leaving the engine.

The nice part of a turbo system is that when not driven hard, mpg is like a NA engine. So, your answer is yes, the turbo spins faster when your foot pushes down on the go pedal. At cruise or down hill or slowing, there is no boost, just like idling.

For turbo longevity, do not drive 'hard' just before engine is turned off. The turbo needs about 1 minute of slow spooling to cool off. If turned off hot, the rubber seals can deform and oil can bypass the seals.
Always use full synthetic engine oil.

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Keeping it running is better than buying new

"...if I accelerate comfortably does the turbo kick in? Or does it only function during rapid acceleration? "

Aftermarket turbo boost gauges are available. The gauge will indicate when turbo is in boost and when not.

Yes, the turbo will boost in rapid acceleration or for increased power demands. It's possible to stay out of boost during a steady cruising speed and a deft technique, but sooner or later boost will kick in when needed. A boost gauge will indicate when turbo boost has engaged.

Keep an eye on the condition of the turbo hoses. They are almost 20 years old.
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Will I buy another Volvo??? We'll see....

Using a synthetic oil of quality is one thing to do. An oil with an ACEA rating of A3/B3 or better is what you want. Mobil1 0W-40 European Formula is one example.
How turbo boost kicks in varies.