Do a search here for terms & conditions - there's been a load of threads lately. My understanding is you can be based anywhere close to a main airport (sorry, don't know the definition of 'main', but LHR applies) as you will be positioned to the aircraft.
Yes, there are many UK based pilots.
If you have an EU passport, you can live and work in any other EU country. If you're not a citizen of an EU country, you need a work permit, which is normally only valid for the issuing country.