(1) They do - as I said, many US airparks for example do not permit landing without permission. I doubt that the proportion is much different betwee UK and US if you add up all runways.
I would expect the proportion to be vastly different.
It's worth drawing a distinction between smaller airfields and larger airports. To extend the analogy above, if I were wanting to go somewhere and park on an acquaintance's drive, I'd call them to ask for permission. But do I need to call Waitrose and ask for permission if I want to go and park in their car park to shop there?
I can quite understand small, private strips, sometimes with planning restrictions, with no AIP entry, requiring PPR. But... Cranfield? Manston? Bournemouth? Duxford? All of these are quite capable of publishing any procedures and notifying changes via NOTAM. Why the need to make use of these airports so inflexible?