Publication rights to a book reside in the book itself, not in the author—although the author will (usually) own the rights to the books that they write.
Once a book appears in print, the first print rights to it have been used up and cannot be retrieved under any circumstances. While other rights to that book might still be available, those first rights are gone for good and nothing, including a change of name for the book or for the author, can legally renew those rights. It's important for writers to understand this as while some publishers will consider republishing a previously-published book, many will not. Self-publishers: take note.
If someone sells first rights to a book and then writes a second book, that second book will have its own full set of rights which the author will own (unless it’s been written under a work-for-hire contract, but that’s a whole new kettle of fish), including first rights.
If you sell full world rights to a book, you can't then sell that book into any other territories or in any other formats, but the publisher which bought those rights can. So if a UK publisher buys first print rights for the UK, US first rights and world electronic rights remain with the writer; if a UK publisher buys world rights to that first publication, they can publish it in their own territory and get their US branch to publish there, for example, or sell those rights to a different publisher. Generally, a writer would get 50% of the advance and royalties resulting from any such sale, but a lot depends on the contract.
Some agents prefer to only sell the rights that a publisher is definitely going to use, which means that the writer retains all other rights: but I think that the best place for rights to belong is with the person or business with the best chance of selling them. There's little point in a writer keeping hold of US rights if he or she has no hope of selling the book into America, when the publisher of their UK edition has a US branch too, or good sales contacts in that country.
Finally, if a book which has already been published is revised, extended, redesigned, and republished, then the resulting book would be a new edition of an old title, not a whole new book—because the main portion of the book would have been published in the earlier edition, so it can't be considered a completely different book.