At this point Arwen has already given up her immortality, but now she grants the grace of the Eldar to Frodo, not giving him immortality since he is not one of the elves, but instead granting him a home in the undying lands when he does pass.
So, No she did not give it to Frodo. Immortality could be given up, but not passed to another. She gave up her immortality to marry Aragorn, who was now the king of both Gondor, and Arnor (The reunited kingdom).
Did Arwen lose her immortality? The answer is yes.
So there we have it, definitive proof from Tolkein himself that Frodo, and his other mortal counterparts, did eventually perish in The Undying Lands.
Princess Arwen Evenstar gave her beautiful necklace to Aragon to pledge her eternal love to him. This special gift from an immortal elf to a mortal man symbolized Arwen's decision to forsake her immortality to be with the person she truly loves.
Arwen didn't want Aragorn to die, because if he died so would she. She said she wasn't tired of living yet. But instead, Aragorn chose to die and Arwen lived a year longer before dying herself.
Once an elf engages in reciprocated love with someone, that's it. If he cannot be with that person he will lose the will to live, die, and become a forever-shade in the Halls of Mandos. Elves do not move on.
Since Frodo saved all of Middle Earth, he is granted access on the last ship to leave Middle Earth in place of Arwen, never to return.
Samwise Gamgee
Because he was also a Ring-bearer, he was allowed to pass over the Sea to be reunited with Frodo in the Undying Lands.
Frodo and Bilbo were comfortable and well off until T.A. 3001. At this time, Bilbo threw an enormous party to celebrate his 111th birthday, and Frodo's 33rd, the date of Frodo's coming of age. At this party Bilbo gave his farewell speech, and made his long-planned "disappearance" and withdrawal from the Shire.
And after their marriage, both Aragorn and Arwen are mortal. Arwen, therefore, could not set foot in the Undying Lands without special permission. And she wouldn't be likely to go after it in the end. So unfortunately, she would not go there, unlike Legolas, after Aragorn's passing.
Arwen reciprocated Aragorn's love, and on the mound of Cerin Amroth they committed themselves to marrying each other. In making that choice, Arwen gave up the Elvish immortality available to her as a daughter of Elrond, and agreed to remain in Middle-earth instead of travelling to the Undying Lands.
According to 'The Tale of Aragorn and Arwen', from Appendix A of LotR, Aragorn II (Elessar) and Arwen had at least three children: one son, Eldarion, who succeeded his father as king, and at least two daughters, as there is a mention of unnamed 'daughters'.
However, he does not possess the ability to live forever, and he cannot choose to be mortal or immortal like Arwen, the half-Elf he loves and eventually claims as his wife. Because Aragorn is mortal and Arwen chose to remain in Gondor and become mortal, their son, Eldarion, is mortal as well, despite his Elven lineage.
Originally, Tolkien intended for Éowyn to marry Aragorn. Later, however, he decided against it because Aragorn was "too old and lordly and grim". He considered making Éowyn the twin sister of Éomund, and having her die "to avenge or save Théoden".
Legolas and Arwen in Rivendell
These two elves aren't actually seen together at any point in the trilogy, but they do in fact have a strong sense of kinship, which can be seen in one of these unused clips where Legolas arrives in Rivendell and Arwen and a host of other elves are there to meet him.
As for Sam, his resistance likely has to do with his innate goodness and loyalty. His love for Frodo helped him to overcome the pull of the Ring and his inner pureness was never fully pierced by the Ring.
Though the Undying Lands can be viewed as heaven, going there doesn't mean death. It is a paradise, yes, but it is not necessarily the end. As far as we know, those elves go on to lead beautiful immortal lives in the Undying Lands.
But when watching Frodo and Sam interact, there always seemed to be a lot more there than just friendship, and that's not even counting the master-servant relationship. In the second book, Sam does say he loves Frodo.
Galadriel is supposed to be banned from return because she associate with the kin-slaying in Alqualondë or at least continued on from Aman and then was banned from return.
After Morgoth's defeat at the end of the First Age, elves like Galadriel were pardoned and offered a return to the land they abandoned. Galadriel refused, partly because she believed her people had no reason to need pardoning, but more because she always wanted to witness Middle-earth and rule a kingdom there.
Eventually, Legolas came to Ithilien with some of his people, with his father's leave, to live out his remaining time in Middle-earth helping to restore the devastatedforests of that war-ravaged land.
Legolas does not marry or have any children, based on the appendix of The Lord of the Rings. Regardless of who his mother was, Legolas is the last of his line. When he sails into the West at the beginning of the Fourth Age, he leaves no family members behind.
In the year 121 of the Fourth Age, after Aragorn's death, Arwen died of a broken heart at Cerin Amroth in Lórien, and was buried there one year after the death of Aragorn, to whom she had been wedded for 122 years. She was 2901 years old.
He feels that going to the havens, where she can be free of the suffering of the earthly world, will bring her true peace.