The point about the remnants of Empire is an interesting one, and depends rather on whether Scottish independence is achieved through a secession or a repeal of the Acts of Union (1707 I think).
If the former, the question doesn't arise, Scotland becomes a new country with only a tenuous connection with the pre-1707 Kingdom of Scotland, with none of the shared assets, liabilities, or international memberships.
If it's the latter, you essentially revert to two countries called England (which had already had Wales for ages) and Scotland; everything, every liability, every asset that has been added since Union is up for grabs (e.g. "OK, if you take Northern Ireland, we'll let you keep Faslane until you get Spadeadam back up to scratch..."
Repealing the Acts of Union is largely irrelevent, meaning your latter possibility is not going to happen. The international norm and the international precedent is that the part of a country wishing to remove itself form the whole is NOT the successor state. East Timor, all of the Soviet Union being prime examples. The rump UK could lose its security council seat (though through what mechanism I don't know), and a few other seats, but will almost certainly be considered the successor state.
Obviously there will be negotiations to divide up the debt. Scotland probably will take about 8~9% of it. North Sea oil and gas will be divided and Scotland will probably get about 90% or thereabouts. Although Salmond's oil fund idea will have to pass a gauntlet of oil companies, who won't want to invest heavily unless they get similar terms they get now.
Scotland will face a big problem choosing the currency. They will almost certainly NOT get any seat on the BoE MPC or any such thing. Or if they did the BoE and rUK treasury will stitch it up to be a largely nominal seat. Scotland would be better off going with its own money in my opinion.
I doubt very much that Scotland will be given automatic access as a member of the EU.
I've no idea if Montserrat, Ascension, Bermuda, Pitcairn, etc. had English or British flags planted, before or after 1707, but it's easy to imagine the sovereignty of the sunnier British islands being up for negotiation as well.
Most of these places would find life as independent nations tricky at best. I'm not sure what benefit they would get form switching from one nominal "mother country" to another. And I doubt a new Scotland would want to manage the foreign and defence affairs of, oh lets say the Falklands.
Biting the hand that feeds IT
