(a) Yes.
(b) No, ownership rules apply.
(c) No they can't, but if - say - it was run (in law) as a syndicate of the registered pilots, the pupils could cost-share at 50% of the total cost.
(d) 100% ownership rules problems again.
To build and run a permit aircraft (PFA or BMAA) would (legally speaking) be pretty straightforward, you just can't use it as training hours, and the P1 has to pay their share of the running costs. You are in Oxford - both organisations are within a 30 minute drive of you, why not arrange to go and see them.
Incidentally, I'd stick with something simple, no matter how good the DT dept. - a Skyranger, Escapade, Savannah or Rans are all sub-500hr kits and would probably all do the job well. Organising a lot of people, even well disciplined schoolboys, for a project like this is rather like herding cats, and you want to make life as easy for you as possible.
For training you need a CofA (but can use a government aerodrome instead of a civil licensed one), you don't for FW a/c need to be an FTO. So, for your particular pipedream you might be far better off buying a wreck which had a CofA, and rebuilding it back to CofA standard. Alternatively do something similar with a Type Approved microlight, which gets you off a lot of expensive hooks (licensed airfield, LAME to sign off the rebuild, requirement for certified engine and instruments, etc.) and allow loggable training hours, although will add the extra cost of getting your "staff" FI rated to teach on microlights.
G