Actually there is already a race that looks suspiciously like an
elf/dwarf mix --- Gnomes!

With some Dwarfish build and delight in craft, some Elvish magic and
interest in woodland thingies ...

For an alternative view of the possibility of Elves and Dwarves
getting together consider also the reactions of Gimli and Galadriel.
For those who've missed Tolkien's Lord of the Rings, Gimli the Dwarf
is less than a couple centuries old, ... Galadriel, Queen and to a
great degree creator of the Elven forest realm of Lorien has been on
Middle Earth over 6,000 years, and has already been married for
millennia, but finds much worthiness in Gimli's admiration. They
don't run off and go bouncy bouncy, they had duties, honor and are in
the middle of trying to save the world and have prior commitments, but
one of the highest of high elves does not react with "Ugh! You're
ugly! Ick!", but with courtesy to a Dwarve's request for a strand of
hair as a keepsake. Details in Fellowship of the Ring.