Are the boot and cabin separate containers
No - it's vanishingly rare to find the boot being a separate weather-tight compartment from the main cabin.
would damp in the boot cause damp in the cabin?
Yep, the rear seats are going to inhibit some of the movement of the damp but not all.
As for how to tackle the issue in general - there's multiple places to check:
Rear window seal - make sure the rear window seal is good. This can be checked by recruiting a friend to watch from the inside of the car and pouring water over with a hose pipe.
Boot seal - the boot lid might not be sealing properly. Test with the same method as above (only this time the hapless friend is in the boot)
Blocked drainage channel(s) - there should be drainage channels to cope with water run off from the rear screen, make sure these aren't blocked.
Boot floor - check for corrosion/holes in the boot floor, water can come up through this when you are driving about in wet conditions.
To deal with the current dampness get everything out of the boot, give everything a clean and dry it down with paper towels. You might want to put a light scattering of cat litter in the base of the wheel well to act as a desiccant to help suck any excess moisture out of the air as well.
are these damp spots
Yep, light corrosion as a result of the damp (possibly mold but I'd say corrosion is more likely)
what would be an effective, preferably natural way to move them?
Take the wheel out and clean it like you would a normal wheel on the car (make sure it's thoroughly dry before putting it back) - there isn't going to be a "natural" way of removing those, you've going to have to use some cleaning chemicals; Bilberry would be my first choice - not too harsh and very effective.
Does the rubber on the wheel need to be treated in any way in case its absorbed anything?
No not really, probably worth giving it a clean down with soapy water on general principles but tyre rubber is pretty hardy - it's designed to face much worse when in use.