What demonic rank does azazel have? Why aren’t grigori mentioned in Ars Goetia?
Well, for one, the Goetia is a specific hierarchy. Just like humans have many different hierarchies, so too do spirits (whether this is fact or simply because that is how humans conceptualise the spirit realm is up for debate).
Azazel does not belong to the same mythology and thus is not part of that particular hierarchy.
Well, I still need to know what position he takes up. Is he a king or does he serve under one?
He is his own being.
He is not part of the same hierarchy and thus does NOT have a rank or position. He serves under no one.
Are there different factions?
According to Azazel’s own mythology, he was the leader of the Watchers, a specific group of angels sent to watch over humanity, but who fell in lust with human women. He taught humans how to make weapons and thus was responsible for much bloodshed, so Yahweh sent the Archangel Raphael to bind him.
There is no one “above” Azazel. He does not bow to any supposed superior. He is his own king.
Yes, there are many factions in the Infernal Kingdom but they are not the same, and do not have the same goals.
Think of it like this: There is the king of England, the king of Scotland, and the King of France. Each is an aspect of the whole, but do not serve the same agenda, even though they share the same continent.