Most UK teachers aren't state employees any more: they're employed by academy trusts.
Semantics. All the funding comes from the state.
It is really not semantics, because of the amount of control that academies have over the work the teachers do, and because of their freedom to hire and fire teachers.
Do they have freedom over how much they pay the teachers?
Yes, academies are not bound to the “Blue Book”, and can pay teachers whatever they like under whatever conditions they like (with the exceptions of teachers TUPEd in).