I have always felt Half-Elves work best when you think of them as D&D's "Dunedain." They are descended from Elves, live longer than other Humans, and have certain "Elven" traits.
In Tolkien's legendarium, Half-elves who have an Elven parent are given a choice, and that choice was granted to all of their descendants - to live as mortals, or to live as elf-kind. Elrond chooses Elf-kind, so in D&D terms, "Elrond the Half-Elven" would be...an Elf. His brother Elros chooses mortality, so he'd be a Half-Elf. And Arargorn, who's descended through many generations of interbreeding Half-Elves remains a Half-Elf.
But what about Arwen? Well, her mom is an Elf, but because her father is (technically) one of the Half-Elven, it seems Arwen (and one would assume her brothers) get to make the same choice, and can choose mortality at any time. Arwen lives as an Elf for nearly 2800 years and then...changes her mind. She then lives another approximately 120 years.
If you follow Tolkien's system, any children of Half-Elves would be Half-Elves or Humans, unless their other parent was an Elf, in which case they would get the same choice.