“The Moon counts as a Land Area with a Bubastis-Controlled Gate for all purposes, expect that Control may not be seized by another Faction. No other Gate can be built or moved there, except Yog-Sothoth via Catnapping.” -Cthulhu Wars Omega Rulebook, page  70.

I’m not sure what the grimoires of the Open Path is. I assume you mean the Spellbooks for Opener of the Way, in which case yes, it counts for that. 

I think perhaps you are confusing the neutral Great Old One Azathoth with the Daemon Sultan Faction. Both use a token called an Azathoth Glyph, so it is a bit weird. 

The Azathoth neutral Great Old One, when summoned, roll a die and adds 2 to it. (Or rolls the Azathoth die in the special die pack, but never mind that.) You pay that much power to summon Azathoth. Then, all other players choose a number from 1 to 6. Add all those numbers together, and place the Azathoth Glyph on the Doom Track on the space that equal what everyone choose. (They get Power equal to the number they choose.) That is Azathoth’s Combat, and it goes down as people fight it. Since it is a neutral Great Old One, once it dies, it goes back into the general Pool for anyone to summon.

If you are playing the Daemon Sultan faction, and you’d like to summon Avatar Thesis, you need to set the Azathoth Glyph on the Doom Track from 0 to 8. You pay that much power, then when you want to summon Avatar Antithesis you pay 8 minus where the Azathoth Glyph is at. The Glyph can change when you summon Avatar Thesis again. 

If you are playing with both the Daemon Sultan faction, and with the Azathoth neutral GOO, you’d probably have to make sure everyone can tell the difference between the two Glyph Tokens. The one in the Azathoth Neutral Pack is grey and plastic, and it looks like the one in the Daemon Sultan is a cardboard token, so maybe that will help. “No one knows. Except Azathoth, who cares not.” -Page 122 of the Final Onslaught Omega Rulebook. 

All pictures were taken from the Final Onslaught Omega Rulebook.

Azathoth glyph token

So, in a game with DS, there’s more incentive for all enemy players to take the fall on revealing a 1, but if you’re in an 8+ game it’s a guaranteed surplus.

I would say yes for the Screaming Dead, because it mimics movement, Yes for Ancient Sorcery, because the Serpent Man can go to any Area when it leave the Faction Sheet. No for Avatar, however, because it switches the places of two units.

The way it is worded it seems like it is not optional. If it is optional, then you must decide before you roll as it is a Pre-Battle Spellbook.

Bastet does count as a cat type.

How Bubastus earns doom is they must have at least 1 cat from a specific planet on Earth (or whatever map you play on) to gain 1 Doom. So, if you have 1 Earth Cat, 1 Cat from Mars, and 1 Cat from Saturn on the Map, you gain 3 Doom. But if you have 4 Earth Cats on the Map, you gain only 1 Doom.