Proof that male vampires can reproduce with living females.

This is shows that a male Cyrodiilic vampire has conceived a child with a living female. It could be part of Clavicus Vile's you-get-to-pass-for-living,-mature-men-with-red-eyes-if-you've-fed-recently package along with sunlight tolerance. A Volkihar, Aundae or those Valenwood vampires that look normal except under candlelight might be sterile.

I don't think the falmer are vampires to be honest, but it is an interesting theory. However the Snow Prince may have been one, though there is nothing I know of that support the theory.

the falmer are just albino goblins in reality nothin unique about them.
as for why they look like bats, they live underground in the dark and crawl around in small tunnels.

If they were vampires they would either have passed on the strain to other elves (Molag bal don't care he just wants more slaves) or they would have huge slave pens.

At the most i think the snow prince will be a snow elf from back before the turned into C.H.U.D.s and he will likely have a small army of other pre C.H.U.D. immortal vampires, who are being hunted by the vigilant of stendarr.

If the falmer lived outside they would not do well,I think they would just get hunted down.