[Galadriel]

Beards are given to those of less nobility? No, I don't think so. All the Wizards had beards, and Wizards were (or at least meant to be) a noble race, an order of the Valar. Círdan was a highly respected Elf, and certainly of no less stature than anyone else. And, as someone up there mentioned, Theoden had a beard (I don't think I need to elaborate on this one).
As for Elves being beardless, I think it would have been a symbol of everlasting physical youth (perhaps). A beard is generally taken to be a sign of maturity (let's forget the Dwarves for a moment).

Well. I can't really understand what I wrote, but anyway...