No, because as you just said, a DAMAGE is not a CHECK (that's true for Ranged damage, spell damage, fire damage, and so on... none of these dice benefit from bonus to checks).
Putting aside the misleading word "damage"... yes, if a spell lets you use your Arcane skill for a check (e.g. using Lightning Touch for a combat check), Acolyte can add 1d4 to that check.
Correct, Ezren will roll 1d12 + 2 + 2d4 when he plays Force Missile. He CAN add another 1d4 with Acolyte, and when he ticks skill feats on his Intelligence later, he will add those too.
The differences between 'die' and 'skill' were tidied up a bit before the release of S&S and the class decks; see this Paizo blog post for an overview, and compare the two versions of Force Missile. Note that in spite of the wording and layout changes, they work the same.
Frencois is saying that if something said it dealt "arcane damage" then Acolyte could not affect it, which is correct.
Nefrubyr more correctly interpreted your question, and stated that the Acolyte could help with arcane combat checks (which many would think of as doing arcane damage), which is also correct.
So, if a card says, "before you act, you take d4-1 arcane damage" then the acolyte can't interact with it. Not that anything does "arcane damage" or anyone would think that the acolyte could interact with damage you take, but still.
And if you play Force Missle, Lightning Touch, Scorching Ray, Disintegrate, etc, you can use the Acolyte to improve your combat check, as the skill you are using is your arcane skill.
Related question. Can Fox (who adds 1d4 to your intelligence or wisdom check) buff Ezren's arcane check with a Force Missile?
My first inclination is to say 'no', because it's an arcane check, not an intelligence check. But I have to ask, one of my players has a bee in his bonnet about this.
Related question. Can Fox (who adds 1d4 to your intelligence or wisdom check) buff Ezren's arcane check with a Force Missile?
My first inclination is to say 'no', because it's an arcane check, not an intelligence check. But I have to ask, one of my players has a bee in his bonnet about this.
Sorry, but I'm pretty sure your player is right on this one. If his Arcane skill uses intelligence (i.e. Ezren's Arcane: Intelligence+2), then it is both an Intelligence and Arcane check. I'm sure Hawkmoon will be by shortly with the relevant rule (if he hasn't already)
Related question. Can Fox (who adds 1d4 to your intelligence or wisdom check) buff Ezren's arcane check with a Force Missile?
My first inclination is to say 'no', because it's an arcane check, not an intelligence check. But I have to ask, one of my players has a bee in his bonnet about this.
Yes he can, assuming it doesn't say "non-combat check." It's the same reason that you can, say, use Blessing of Gorum on a fighter using his Melee (strength+x) for a weapon. Ezren is making an Intelligence and Arcane and Combat check (and also Attack, and Force, and perhaps more).
EDIT: Ninja'd!
An Arcane check based on the Intelligence skill is an Intelligence check. See this FAQ. I know both isaic and Orbis answered it, but I'm just pointing to the FAQ that seals the deal.
So Ezren's Arcane skill can be buffed by a Fox, but Seoni's (which is based on Charisma) cannot.
I agree with all that.
Note though that, while the Fox can't help Ezren with a Wand of Force Missile, the Acolyte can. The trait on the cards also determine the type of check, and the Wand of Force Missile allows Ezren to make an Arcane check that isn't related to his Intelligence.
He won't get the Intelligence die, Intelligence skill feats, or his Arcane bonus modifier (because only gets that when he uses his Arcane skill). But it is still his Arcane check, which the Acolyte would help with.