If I actually go for writing more of the fic, I'll probably pull a Mike Mignola and shuffle the demon/hell stuff and Lovecraftian stuff together a la the first Hellboy movie, so the monsterous tentacled things would be on the YED's side (they occupy the same Hell dimension Lucifer does, so freeing Lucifer lets them out as well). Mostly, I just wanted an excuse to have (non-sexual) tentacles in there.

Veidt I think summons the YED to make a deal with him in order to avert the nuclear armageddon he believes is coming. Sadly for him, he didn't realize that when Azazel said, "Make a deal with me and I'll avert nuclear armageddon" he meant "By causing actual armagedon." (probably he thought he could control him, but of course couldn't, because you never come out ahead in a deal with a demon -- that's not how the genre works).