have a governor-general as her representative, and they are sometimes referred to as her "realms". I suppose that if the Queen went to live in the Bahamas, the UK would have to appoint a governor-general.
Probably not, the UK would probably appoint Councillors of State under the Regency Act to perform the Royal functions. This is the practice when the Queen is absent from the realm. An alternative would be to appoint a Regent under the Act but the conditions for that would probably not be satisfied.