Arwen didn't want
In the same text, Tolkien wrote that Eldarion had lived until the age of 125, most likely having passed away in Fo.
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.
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.
Elrond has only Half-Elf, and he was once given the choice to be counted among Man or Elves. When he chose to be counted as Elven kind, he was then given immortality. Because Arwen is Elrond's daughter and also Half-Elf, she too has the choice to be a mortal woman or an immortal Elf.
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'.
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.
Her constitution already severely weakened by loneliness and despair, combined with physical injuries, Éowyn then succumbed to the Black Breath. Théoden, not knowing that she was nearby, told Merry with his dying words that she had been "dearer than daughter" to him.
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.
Partially this was due to the unique situation of their ancestry: because of the complications of being descendants of marriages between mortals and immortals, Elrond and his brother Elros were given the choice of their own fates, to be mortal with humanity or immortal with the elves.
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".
So by Aragorn living 210 years, he was given a lifespan of a king who lived at a time when the Dúnedain were still strong. The Chieftains previous to Aragorn had been living into their 150s, except for those who had died an untimely death (usually in battle).
Why was Arwen's fate tied to the ring? Because Aragorn's fate was tied to the Ring. Aragorn, remember, was not going to be King of Arnor and Gondor UNLESS Sauron was defeated. Sauron could not be defeated by any other means than destroying the Ring.
She has a vision of a child clothed in white running across her path. She watches as the boy runs with his arms outstretched to his father, an aged Aragorn, who picks him up with joy and kisses him. As the boy looks at her and she sees he is wearing the Evenstar, she knows the boy to be her child.
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.
Average Elven Lifespan
Elves have the ability to live forever thanks to their immortality. Elves like Elrond, who was over 6000 years old in The Lord of the Rings, are not hard to come by.
Arwen was her granddaughter
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).
Because Arwen has forfeiten her elven privilege to board the ships to the west a long time ago. When Arwen died, she died as a human foregoing her Elven self.
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.
Éowyn and Faramir were married, and dwelt together in the hills of Emyn Arnen in Ithilien. She kept her friendship with Merry, and when he became Master of Buckland in Fo. A. 11, she sent him great gifts, as did her brother from Rohan.
Just no. He'd been engaged to Arwen since literally before Eowyn was born.
Yes. They had one son, Eldarion, who succeeded his father as second King of the Reunited Kingdom of Gondor and Arnor. They had an unspecified number of unnamed daughters as well. Eldarion's birthdate is not given, nor is the birth order of any of their children.
Because she has both Elven and Human blood she has a choice. She dies eventually after mourning Aragorn for a long time. She leaves Minas Tirith and lives alone in fading Lothlorien for a while before dying. This is all detailed at the end of RotK, in the appendices.
At the end of The Lord of the Rings, Frodo sets sail for the Undying Lands, which implies that he will live in peace and harmony for all eternity. This is where the immortal Elves come from (most recently) and where they return to when they find that their work in Middle-earth is done.
And Tolkein himself confirms this in his own private letters, where he makes it clear that Frodo and his friend Sam (who is also eventually granted a place in Valinor) would always be mere mortals and would eventually succumb to death even if they were to remain in The Undying Lands.