If someone tries to hit a shadow created by the Project Image spell, does someone have to roll to hit? If so, does it have a low AC like a Figment, before they roll Will to disbelieve, or does it work differently?
[RPG] the AC of a projected image
First off I want to congratulate the Wizard for coming up with such an innovative plan for stopping a dragon.
All Spell DCs will be based on your Wizard's primary spell DC
So, at base, this is 10 + Spell Level + Primary Casting Attribute Modifier (Int for the wizard). Assuming an Int bonus of +4, the save DC on Mind Fog is going to be 19.
Failing that first save is going to make the following a lot harder:
There is not a modifier for the disbelief save
The DC is going to be 10 + 3 + Casting Attribute.
But your baddie must interact with the Illusion first, before getting a save. So for Cthulhu, it would mean probably coming into contact with the illusion physically. Now a Dragon is pretty smart, and probably going to think that it's an illusion (especially if it doesn't think someone is dumb enough to summon an Outsider to stop it). It could charge Cthulhu head on if it's brash (and actually assumes it's an illusion). Though, safer ways to go about that would be to try Detect Magic and on round 3, you can determine the school of magic. Or even faster is True Seeing which will get past the illusion instantly.
So, your Wizard will not want to have Cthulhu actually interact with the Dragon, otherwise it'll trigger the save.
It should be noted that the Dragon will probably make it's spellcraft roll to identify Black Tentacles, but it may or may not assume that is being caused by the weird outsider and not some illusionist trying to save the King.
To address the Ghost Sound + Silent Image save DC
You'd save against them separately at their different DCs. Though, if you succeed on the save for Ghost Sound, I would count that as "interacting" with the Silent Image, since you will realize that the sound the illusion was making is an illusion.
Edit: As @SimonGill points out, the enemy army would most definitely freak out seeing this, and would probably not get their disbelief save...
To the best of my knowledge, the issue is never directly addressed. The rules, on a quick scan, appear the same as in 3.5, where this issue has been debated some as well.
The long and short of the argument is this: the statement in shadow evocation et al. is that someone who knows that it is fake does not need to save. Nothing says he does not save or may not save, just that he doesn’t have to. This is taken to mean that this is optional, and effectively someone in this position has the option of automatically succeeding on his save the same way you typically have the option of failing any save. You may, according to this logic, choose not to automatically succeed, and then, since you are now attempting a saving throw, choose to automatically fail.
Strict-RAW, this seems most accurate, though it definitely takes a few steps to get there and it’s clearly not written out explicitly. Still, the language, whether it was intended to be or not, is precise: it waives a requirement to save, it does not add a requirement to not-save.
Whether or not you should rule this way in your game is more dubious. Shadow evocation et al. are rather useful, particularly for this feature. In 3.5, greater shadow evocation was typically used to cover the loss of contingency due to the banning of Evocation as a specialist wizard. In Pathfinder, this is less of an issue (since banning is no longer so absolute anyway). Most of the time, shadow evocation et al. are most useful when the drawbacks of using them (the Will save, the quasi-reality) don’t actually affect the functioning of the spell, which is precisely in this case: buffing. Ultimately, it becomes yet another powerful and flexible tool in the wizard’s toolbox, and he’s already got a ton of those. Shadow evocation et al. are’t the most powerful of them, but maybe it’s worthwhile to you to start paring down options where you can.
I would say it has the same AC as the caster.
It's not a figment, so the figment AC rules don't apply. There is no general shadow AC rule - other specific spells have them, but that's not really relevant.
It's a seventh fricking level spell. A seventh level spell to make an illusion of you - I'd give the PC a hell of a lot of leeway on it. Otherwise it seems like you could do the same thing with much lower level illusion spells.