So does this mean an unrestricted instructor may hire out an aircraft from, let's say, a private owner as opposed to a business and then teach someone NPPL? No office for ground training, literally just an aircraft?
Legally yes, there is nothing to prohibit you operating from the boot of a car however, the aerodrome operator may have something to say about it. It doesn't matter who owns the aircraft so long as it is maintained to public transport standards if the student pays to use it.
The instructor does not have to be insured, but the aircraft does.
If the instructor isn't operating a RF does this mean they would need an air operators certificate?
No, because you cannot give instruction under an AOC; its for commercial air transport. The instructor and student have to both be members of a club for which there is no legal definition.
Would any of the hours conducted under the NPPL perhaps count towards a full JAR licence under these circumstances?
A maximum of 10 hours. All JAA training must be conducted at a Registered Facility or FTO.
Can any JAA FI/JAA FE teach/examine an NPPL?
Legally Yes. However a JAA examiner would have to be Authorised by the UK CAA, not just any JAA Examiner.