Can they legally fly if designed/modified and built by an amateur
Yes. Simple mods are overseen by the PFA, major mods by the CAA. All mods have to comply with BCAR Section T.
and if not, what constitutes an amateur?
Usual PFA / CAA interpretation - ie done for recreation rather than profit.
Does an autogyro/gyrocopter require a Permit to Fly, who issues these and how is the machine deemed airworthy and safe?
Yes - issued by the PFA or the CAA. Airworthiness and safety are measured through compliance with BCAR Section T.
If a test pilot flies it, is he independent from the builder?
It can be the builder if he/she has appropriate experience. The CAA currently also require an independent test by their own nominated test pilot.
What if he decides it’s safe enough for an experienced pilot but not for an inexperienced pilot? Does he still issue a Permit or would there be restrictions on it?
If that's the case then it wouldn't meet the requirements of BCAR Section T, which require the aircraft to be controllable by Captain Average. Some current designs which predate BCAR Section T and do not meet its requirements have minimum pilot experience requirements imposed.
Are there any rules that are different from fixed wing “home-build and design”?
Yes - gyro pilots require gyro licences. Gyros are approved against BCAR Section T rather than Section S (microlights) or JAR VLA (or whatever it is) for fixed wing.
Happy now Bronx?