NO. Frodo did not become immortal. Arwen essentially gave up her mortality, when she plighted her troth to Aragorn; because she KNEW as a fact, that he would defeat Sauron. At that point, she gave up her right to return to the Undying Lands.
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.
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.
Galadriel and Celeborn had a daughter named Celebrian, who married Elrond and became the mother of Arwen (as well as her brothers, Elladan and Elrohir).
Arwen. Galadriel's granddaughter and the daughter of Celebrían and Elrond. After the War of the Ring, Arwen marries Aragorn II Elessar and becomes Queen of the Reunited Kingdom of Arnor and Gondor. Like other Half-Elves, Arwen has the birthright to choose either immortality among the Elves or mortality among Men.
She was the granddaughter of Galadriel and Celeborn and often called Arwen Undómiel or "Evenstar". Through Aragorn, Arwen was the adoptive sister-in-law of Legolas Greenleaf and the paternal adoptive aunt of their children: Anna, Nessa, Thaulius, and Nellriel Telcontar.
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.
Elrond is telling Aragorn that because she has chosen him, she is now slowly dying the death of any mortal, as they live and grow older. Seeing as (in the movie of course) Aragorn left thinking that Arwen had chosen immortality and was leaving Middle Earth, this is important information for him to know.
Only Elves and those deemed worthy (like Gandalf and Frodo) could sail to the Undying Lands. And by choosing to stay with Aragorn, she was forgoing her place on the boats and binding herself to the realm of men, thus becoming mortal.
In tying Arwen's fate to that of the One Ring, Jackson aimed to show that there is more at stake than just the trees and the innocent creatures, there is also the light of the world that can never be allowed to succumb to the dark.
Men were never allowed to enter the Undying Lands after the fall of Numenor and any attempts to reach the lands would result in failure or disaster. Aragorn, despite his heroic deeds, likely wouldn't have accepted an invitation to ensure his duties as king.
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.
The fantasy and science fiction author Marion Zimmer Bradley wrote that Frodo and Sam have the most intense love described in the book, stating that towards the end they attain "classical 'idealized friendship'" of an emotional force like that of Achilles and Patroclus in the Iliad, or David and Jonathan in the Old ...
After the downfall of Sauron, Gimli travels with many of his kinsfolk to the Glittering Caves of Aglarond. He becomes the Lord of the Glittering Caves, and prospers there for many years, doing great works for the Kings of Gondor and Rohan. He never takes a wife and has no children.
Celebrían was an Elf-lady of Lothlórien, the daughter of Celeborn and Galadriel. As the wife of Elrond, she was also known as Lady of Rivendell. She was the mother of Arwen and the twins Elrohir and Elladan.
Like Elrond, some of Galadriel's powers stemmed from the Ring of Power that she held in her possession. Nenya, the ring of water, had the powers of preservation and concealment from evil, which Galadriel used to create Lothlórien and keep it safe from outside influences.
The direct descendant of Elros is Aragorn. In terms of genealogy, Elros would be Aragorn's great grandfather with about 60 "greats." This makes Elrond Aragorn's great, great, etc. uncle, and since Arwen is the Lord of Rivendell's daughter, she and Aragorn are first cousins many times removed.
When The Fellowship is taken to Galadriel, she and Boromir have a telepathic conversation which immediately causes Boromir to burst into tears. She senses his desires for the Ring but tells him not to give in because even now, there is hope.
Before the Company of the Ring left Lothlórien, each of its members was presented with a gift by Galadriel. Merry and Pippin were given "small silver belts, each with a clasp wrought like a golden flower".
