Welcome, TheLostPilgrim!
You bring a lot of good questions, let's see if I can avoid being too confusing.
The Grey Havens were merely a harbour maintained by Círdan and the Elves with him as a point of departure for Elves in Middle-earth to return to the Undying Lands.
The Havens themselves were really nothing special, and they would have been accessible to anyone. They were guarded by the Elves though, so unauthorised entry was not allowed.
As for the Blessed Realm itself, it seems clear that mortals going there by some grace did not acquire immortality. I don't know if you've read The Silmarillion, but there it is said that the life-potential of the "speaking peoples" of the world was the sole province of the One who'd made them, and could not be permanently altered by any of his other creations without his approval. Bilbo, Frodo, Sam, and Gimli would, at the end of their natural lifespan, die.
The immortals like Elrond and Galadriel, would fulfill their own potential as Elves, living as long as the world itself survived.
Olórin was, for all intents and purposes, Gandalf in personality and bearing. Since the Maia were capable of altering their physical forms at will, I see no reason why he could not have appeared to the Hobbits and Gimli in the "Gandalf" form as long as they lived. He was indeed "eternal", at least as far as he was an "angel", a created being of a higher order than Elves, Men, or Dwarves.
The issue of the Elves is bit complex. They had to leave Middle-earth to make way for the Dominion of Men, an event in the plan of the One from earliest creation. Also, since they were immortal, they could never truly be content in mortal lands, at least not forever. Witness the way in which Legolas, born and raised in the Old Forest far from the Sea, became obsessed with sailing after he encountered it in person.
The Elves weren't forced to leave, though it was made clear to them that if they remained in ME they would ultimately fade to the point of a mean existence in caves and hidden places, forgetting their own natures over time. If any did stay beyond their chance to sail West, that would have to have been their fate.
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Music alone proves the existence of God.
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