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:
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:
She doesn't seem to ever reconcile herself to it.
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.
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.