What is the Sky Forge? It's an ancient magic of some kind, enabling regular mundane steel to be annealed to a strength on par with weaponcrafts of the Elves.

But where did it come from? Why does it have a giant hawk looming over the pit?

Take into consideration an excerpt from http://www.uesp.net/wiki/Skyrim:Songs_of_the_Return,_Vol_7:

Considering that it has a hawk looming over it (and of theological symbolism of the Hawk in TESLore), that the elves are scared of it, and that its magic is "almost as old as Nirn itself"... I think its name might literally describe its first function. I think it is where Kyne forged the sky. She's a very padomaic aedroth, the "War-wife of Shor", so that may also explain why the snow elves were scared of it: it was the lair of one of the strongest of the "enemy" aedra to the elves, and there is mythical remembrance passed on through the generations of Falmer about Her wrath coming down on any who assailed it.

http://steamcommunity.com/sharedfiles/filedetails/?id=15626&searchtext=uncle+sheogorath built it.

we're welcome.

:-)

Neat idea, I like it. Are there any more references to the forge in the in-game lore? It'd be nice to compare all known bits of information and see if it is consistent with this hypothesis.

If the Skyforge is where Kyne forged the sky, is the Lunar Forge where she forged the night?

If anything, the Lunar Forge might be more closely associated with Lorkhan.