Yeah the rubber part gets replaced.
Correct the spare tyre is the wheel+tyre as you can’t just swap the tyre on your own. And yeah rotating is just moving wheel+tyre to a new position on the car so the tyre gets a different wear pattern which helps it last longer.
It is like a bike :). The wheel is the metal/plastic part in the middle. The tyre is the outer rubber part. They will take the wheel of the car to replace the tyre but unless the wheel is also damaged you will get the same one back. Regarding the inner tube some have a tube and some don’t same for bikes actually.
From the British TV show IT Crowd. I forget the exact scene but it is the father trying to convince his son he is in heaven and Hitler ruins it.