Does Frodo live???

[Mankáno]

Daniel, regarding the destiny of men, dwarves of hobbits, it's not said what happens with them, it's said that not even Manwe knows their fate after they die, the thought of them going with Iluvatar is only a presuncion.

Regarding Tuor, again, it's never said that he become inmortal, not even that he was indeed admitted in Valinor or even got there.

The only fact given in the Silmarillion is that he sailed to the west with his wife Idril and nothing else was known of him or her.

Now, it says it is "believed" that he was admitted in Aman but it doesn't say he become inmortal. However it's said he was the only mortal man to be numbered among the Eldar, from that you could conclude that he become and elf himself.

Now, all those if are of course deliberate, it was Tolkien's intention not to be clear in this matter, as the Silmarillion and the LOTR were supposed to be written in Middle Earth, there were no way they could possibly know what happened in Aman after the Noldor left it at the begining of the first age. The few things known, as the Earendil trip are because some elves came with the army and told it to their friends, some of who stayed in Middle Earth for a while.

Now, assuming that Tuor was indeed admitted among the elves and become inmortal, I can only suppose an explanation.

First, the men and elves' fate were determined by Iluvatar, the valar couldn't change it but Iluvatar could. He did it in the case of Luthien and he could, at his will make an exception with Tuor. Why would he do that?

Well, Tuor was a choosen one, predestined to have a very important role, he was protected by Ulmo and maybe Ulmo asked Iluvatar for it and Iluvatar listened.

In addition, Earendil, Elwing and their children, because of their condition of half-elves were given the choice to belong to one race or the other, half-elves weren't in the music and their fate wasn't clear, so, Iluvatar allowed the Valar to make a decision and they gave them the choice, the choice was extended to their descendants and the Valar could have extended it also to their only mortal ascendant who was still alive: Tuor.

this is of course my own interpretation, there isn't anything in Tolkien's work that says that and he probably never thought of it.

Regarding Frodo it is different because there is indeed a letter from Tolkien that says he wasn't admitted for ever but only as a healing process for his soul, for a very long time, but not for ever.