Certainly in the case of the limited company groups of which I have been a member, not all flying group members were registered with the CAA, merely the company itself and nominal directors(s)
Correct.
If the aircraft is maintained to 'public transport standards' then you do not have to be a registered owner to
rent it. By rent, I'm talking about 'non-equity' groups, or flying 'clubs' where your membership entitles you to use the aircraft.
However, if you own a share then you should legally be recorded with the CAA as a registered owner (see the Trustee form link above). If you own a share in an aircraft, but it's registered to someone else's Limited company, you would have virtually no say in the matter if that Director decided to sign ownership of the aircraft away to himself and run away with the money!
And in terms of the 'maintained to public transport standard' definition, unfortunately this is something that we have sought clarity regarding but there doesn't seem to be a straight answer, especially with the EASA transition. Basically, anything with a CofA would be maintained to public transport standards as part of the permit process. However, there is some blurring of the boundaries between this and Permit to Fly aircraft, particularly those that have been built in a factory eg. the Ikarus C42.
To be honest if you have any questions, you should email aircraft.reg(at)caa.co.uk - they deal with the insurance, registration/trustee/ownership issues and will normally reply within a week quite unlike many other CAA departments.