Somewhere in HoME Tolkien has Manwe state that any person with Mortal blood, no matter how much, is automatically subject to the Gift of Men 'unless given another doom'.
The power to choose between the Gift and immortality was a special grace appointed to Earendil and Elwing, their children and Elrond's children for certain. I doubt it would have been extended to Arwen and Aragorn's children. It certainly was not granted to Half-Elven of other lines, (such as the Princes of Dol Amroth).
Dior was thus Mortal, though he undoubtedly would have proved very long lived had he not have been murdered by the Sons of Feanor.
Half-Elven combine the traits and gifts of Men and Elves. The choice effects only their ultimate destiny it does not change their hybrid nature. Thus Arwen would not loose her pointy ears, (assuming she has them) or the powers inherited from her Elven ancestors.
Aragorn, dispite the many generations seperating him from Elros, retains a number of Elvish traits including his power of healing. Though undoubtedly Mortal the same will certainly be true of his children by Arwen.
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