In the beginning Eru, the One, who in the Elvish tongue is named Ilúvatar, made the Ainur of his thought; and they made a great Music before him [...] The Great among those spirits the Elves name the Valar, the Powers of Arda [...] With the Valar came other spirits whose being also began before the World, of the same order as the Valar but as less degree. These are the Maiar, the People of the Valar and their servants and helpers. Their number is not known to the Elves, and few have names in any of the tongues of the Children of Ilúvatar; for though it is otherwise in Aman, in Middle-earth the Maiar have seldom appeared in forms visible to Elves and Men [...] Wisest of the Maiar was Olórin. He too dwelt in Lórien, but his ways took him often to the house of Nienna, and of her he learned pity and patience. [...] Of Olórin that tale [the Quenta Silmarillion] does not speak; for though he loved the Elves, he walked among them unseen, or in forms as one of them, and they did not know whence came the fair visions or the promptings of wisdom that he put into their hearts. In later days he was the friend of all the Children of Ilúvatar, and took pity on their sorrows; and those who listened to him awoke from despair and put away the imaginations of darkness. - The Silmarillion

"It was these records that brought the Grey Pilgrim to us. I first saw him when I was a child, and he has been twice or thrice since then."
"The Grey Pilgrim?" said Frodo. "Had he a name?"
"Mithrandir we called him in elf-fashion," said Faramir, "and he was content. Many are my names in many countries, he said. Mithrandir among the Elves, Tharkûn to the Dwarves; Olórin I was in my youth in the West that is forgotten, in the South Incánus, in the North Gandalf; to the East I go not."
The Two Towers: "The Window on the West," p. 278-79

somebody in the tolkein society had a theory that he is mandos in discuise.

(Freebie tidbit: Both Sauron and Saruman are/were Maia also. 8))

Sure people can die there. Even Elves can die there even if they are not killed. Miriel, mother of Feanor did.

Rain-maker, when Frodo destroyed the Ring, he destroyed a ton of Sauron's power - enough that now Sauron can never take shape again to do great harm.

Actually, people DO die in the Undying Lands. The name means that the land itself can never "die" of age. It will always look the same as it did at the beginning of time. I remember reading this in ROTK.