Do Elves Get Cold?

Elves do seem to have more resistance to cold– c.f. Legolas wearing only “light shoes” on Caradhras; I believe that's what started the “fanon” idea that they don't feel cold at all, under any circumstances– which is certainly not the case, as you've just demonstrated.

Are Elves immune to cold?

All High Elves, barring very rare circumstances, live forever; Wood Elves, although lacking immortality, still live well into their seventh or eighth century . Elves cannot be harmed by natural disease, are incredibly resilient to heat and cold, and never age past their prime.

Can Elf get sick?

First, off, the elves never get sick. It's one of the benefits of being an elf (they can, however, be affected by strong poisons.) So, when discussing illness in Middle Earth, keep in mind that these illnesses would only affect mortals.