[Finwe]

I think that Galadriel was able to go to Aman because she had done her "penance" by refusing the lure of the One Ring. I think the Valar were mainly testing her to see if she had merely escaped the main bloodshed in the First Age by chance, or by choice. If it had been by chance, then she wouldn't have refused the One Ring, being proud and mighty like most of Finwë's line. If she had escaped the bloodshed by choice, then she would refuse the One Ring, because she had been making conscious decisions to not involve herself in the same bloodshed that killed her kin. It was a test to see if she truly was "pure" enough to go to Aman.

Nenya, being the Ring of Water, increased the Sea-longing in Galadriel when she bore it. In the Unfinished Tales, it is said that once, the Sea-longing became so great in her, that she "moved" to Belfalas, and lived there for a while. Even though that seems to be a one-time thing, I think that Sea-longing remained in her. The only things that kept her in Middle-earth were: 1) She knew that she had to pass her "test," and until Frodo came to Lothlorien, she knew that her test hadn't come; and 2) She needed to remain and defend Lothlorien until the Enemy was finally vanquished. Her desire for the Sea was overcome by her sense of duty toward the people of Middle-earth, and that is why she stayed as long as she did, and why she left when she did. At the end of the Third Age, Sauron was vanquished, Aragorn made King, and peace brought to most of Middle-earth. Galadriel knew that she (and the Elves) were not needed any more, and that the Dominion of Men were coming. She knew that the Elven kingdoms/territories would interfere with the Dominion of Men, and thus, she left (along with Elrond, the leader of the other main Elven community).