I am sure Elrond is half-elven, so Arwen is too. I recall something saying Argorn was half-elven as well. Shouldn't half-elves not have to become mortal to marry each other?
Saying that someone is "half-elf" is a way of saying that they have an elven parent and a mortal one. Elrond fits into this category (his mother, Elwing, was an elf; his father Earendil was a human). But one is not thereby "half-human, half-elf" - one has to choose between being an elf and being a human:
"... Elrond Half-elven, who chose, as was granted to him, to be numbered among the Eldar; but Elros his brother chose to abide with Men." (Silmarillion chapter 34; page 254 in my edition)
Thus, at the time of the War of the Ring, Elrond is quite definitely an elf. Arwen gets that same choice.
But Aragorn is quite definitely a human; he's a descendant of Elros. Thus, he is going to die; and Arwen makes the choice to die as well, that is, to be under the Doom of Men, in order to be with him. It is only at the very end of her life that she sees what's involved with this and regrets it:
"If this is indeed, as the Eldar say, the gift of the One to Men, it is bitter to receive." (Lord of the Rings, Appendix A, p. 1038 in my edition)
She doesn't seem to ever reconcile herself to it.
I doubt she had to; rather she decided to.
There's precedent for a full-blooded (immortal) Elf marrying a mortal, where the Elf retains (or seems to retain) their immortality:
- Idril (daughter of Turgon, King of Gondolin) and Tuor1
- Mithrellas and Imrazôr
There doesn't appear to be any requirement that an immortal must become mortal in order to marry a mortal; in fact, they can't. Lúthien is the only full-blooded Elf who did this, and she had to die and be reborn. Only half-Elves of the line of Eärendil were given the mortal choice.
Having said that, by all accounts Arwen decided to give up immortality; bear in mind the words she uses when professing her love to Aragorn:
I will cleave to you, Dúnadan, and turn from the Twilight.
Return of the King Appendix A: "Annals of the Kings and Rulers"
And what Tolkien writes in Letter 153:
When she weds Aragorn (whose love-story elsewhere recounted is not here central and only occasionally referred to) she 'makes the choice of Lúthien'
The Letters of J.R.R. Tolkien 153: To Peter Hastings (draft). September 1954
Notably, he frames her choice in terms of Lúthien, not in terms of mortality/immortality. I'll get to why I think this is important in a moment.
The question then becomes: why did she make the choice?
Tolkien's remark above gives one clue; "the choice of Lúthien" is described thus in The Silmarillion:
Because of her labours and her sorrow, she should be released from Mandos, and go to Valimar, there to dwell until the world's end among the Valar, forgetting all griefs that her life had known. Thither Beren could not come. For it was not permitted to the Valar to withhold Death from him, which is the gift of Ilúvatar to Men. But the other choice was this: that she might return to Middle-earth, and take with her Beren, there to dwell again, but without certitude of life or joy. Then she would become mortal, land subject to a second death, even as he; and ere long she would leave the world for ever, and her beauty become only a memory in song.
The Silmarillion III Quenta Silmarillion Chapter 19: "Of Beren and Lúthien"
If we take the Professor at his word, then one of Arwen's motives is revealed: she's so in love that she would rather die than spend the rest of eternity without him.
As well, though there's no evidence she was thinking along these lines, I've commented before that the nature of the Eldar is such that there are practical benefits to not marrying beneath your mortality:
[T]he life and love of the Eldar dwells much in memory; and we (if not ye) would rather have a memory that is fair but unfinished than one that goes on to a grievous end.
History of Middle-earth X Morgoth's Ring Part 4: "Athrabeth Finrod ah Andreth"
But that's perhaps reaching for explanations; I tend to agree with Matt Gutting that she wasn't thinking that far ahead.
1 There's actually in-universe speculation that Tuor was granted immortality, though personally I find it doubtful; I'm particularly intrigued by Tolkien's statement in Letter 153 that "'it is supposed' (not stated) that he as an unique exception receives the Elvish limited 'immortality'"
As half Elves, Elrond and his brother Elros had to choose between being mortal men or being immortal Elves. Elros became a man, Elrond became an Elf.
But to the children of Elrond a choice was also appointed: to pass with him from the circles of the world; or if they remained, to become mortal and die in Middle-earth. For Elrond, therefore, all the chances of the War of the Ring were fraught with sorrow.
-The Return of the King, Appendix A, "The Numenorean Kings"
So it isn't that loving Aragorn itself requires Arwen to become a mortal; staying in Middle-earth after Elrond departs will make her become a mortal.
As for Aragorn being a half-Elf, he isn't. See the answer to this question to see just how little Elven blood ran through Aragorn's veins. He was something like 0.00000000000000000677% Elf.