On the subject of famous weapons, let us remember Grond and Ringil
"Then Morgoth hurled aloft Grond, the Hammer of the Underworld, and swung it down like a bolt of thunder"
That's one of my favourite quotes from the Silmarillion

That's one problem I have found with bookstores in the US. They NEVER have the Histories of Middle Earth. Even if you go online and try to order it, all they seem to have is volumes 1-5. And I might add, these are very unimpressive volumes. I remember seeing them all, in 3 large black volumes in Waterstones (the UK's finest bookstore), and I wish I had had the money to buy them there and then... :thumbsdow
Anyway, back on topic!

On Frodo and Bilbo:

They were indeed not granted immortality, for who in Valinor had such power? Not even the Valar could do that. Remember Ar-Pharazon's arrogance and mistake when he sought to seize immortality for himself from the Valar by invading Valinor in the Second Age? Neither the Valar or the land of Valinor itself could grant immortality. Those who dwelt there were immortal because they were simply made that way... it was in their nature. Mortals were e xp ressly forbidden to go there, but even if they did, that would not make them immortal.

Thus, Frodo and Bilbo being mere hobbits from Middle-earth, they were quite mortal by nature, and thus would eventually die of old age, regardless of where they spent their last days. In fact, Bilbo and Frodo did not go to Valinor, they went to the Lonely Isle, Tol Eressea, which though a part of the Undying Lands, was not mainland Valinor itself. I suspect that their normal lifespans were lengthened considerably by living in those hallowed lands, but they still died.

Bilbo, Frodo (and Sam) would eventually die in the West. As Valkrist pointed out, in spite of the moniker "Undying Lands," it is not the land that confers immortality--it is the essential nature of the beings who happen to live there. Although there are exceptions...Feanor's mother died of what sounded like turbo post partum depression and refused to take on incarnate form a second time (as is an Elf's right/duty). Getting back to the Hobbits, only Iluvatar (God) could change their nature and make them immortal. The purpose of going to the West in spite of the fact that they will still die it to gain healing from the terrible psychological wounds caused by the Ring. Only there could they gain such healing, and because of their importance in the overthrow of Sauron, they were granted the boon to pass into the West and find peace before death, before passing through the Halls of Mandos into the unknown beyond.

Where would they live? I don't remember for sure, but the Elves lived primarily on Tol Eressea, though they could visit Valinor for feasts and whatnot. There was one house of the Elves (the Vanyar?) who were so goody-two-shoes, er, noble, that they dwelt at Manwe's feet (presumably playing harps 24-7) on Taniquetil. I'm guessing that the Hobbits would have lived on Tol Eressea, but could have visited Valinor from time to time.

Morgoth and Sauron.... Morgoth was the most inherently powerful creature in all of creation. However, he instilled his evil essence into the fabric of Middle-earth and couldn't necessarily draw on his full might. However, the fact that some of his power passed into the world accounts for the continuing sorrows of the Children of Iluvatar even to this day--a prime case of "gone but not forgotten." Sauron, on the other hand, was IMHO a much more successful Dark Lord, in spite of being inherently weaker. He survived two ages of Middle-earth, compared to Morgoth's one (though we don't know how long Morgoth was around stirring up trouble in the time before the First Age officially began [though some of that time was spent chained up]). Morgoth ultimately overran all of Beleriand during his tenure, but Sauron was apparently undisputed lord of all of Middle-earth east and south of the part we're familiar with. (Tolkien stated somewhere that Sauron's nominal capital, as it were, was in the east, and Mordor was established as a barrier against the Dunedain kingdoms.) And Sauron brought about the utter destruction of Numenor, the most powerful human kingdom of those days (though not without considerable cost to himself).

That's my take on these issues, anyway.