Yes, you must choose one of the listed skills.
For example, if a check lists Dexterity, Disable, Strength, and Melee, you may use any one of those skills to attempt your check.
So if you choose Intelligence as the skill you will attempt, ignore completely the fact that Arcane was a possibility. Thus Kyra is making an Intelligence check, so the Acolyte will not help her.
Ezren is making an Arcane check, ignoring that Intelligence was an option on the card he is attempting to acquire. But what he doesn't ignore is that his Arcane refers to his Intelligence. So if he boosts his Intelligence, he is boosting his Arcane as well.
If, on your character card, the skill you’re using refers to another skill, both skills count for the purpose of determining the type of check. For example, if you’re using the Arcane skill on a combat check, and your character card says that your Arcane skill is Intelligence +2, the check counts as both a combat Arcane check and a combat Intelligence check.
So for Ezren, his Arcane check counts as an Intelligence check. But for Kyra, it doesn't. So Ezren could play the Sage or the Acolyte to help him attempt an Arcane check to acquire a spell. If Ezren for some reason chose to attempt an Intelligence check, he couldn't use the Acolyte.
If Kyra is making an Intelligence check, she can use the Sage. If she makes an Arcane check, she can use the Acolyte, but not the Sage. Kyra's Arcane skill (the unlisted d4) isn't based on her Intelligence, so boosting her Intelligence doesn't boost her Arcane.
Similarly, Seoni can get help for an Arcane check from an Acolyte, but not from a Sage. Seoni could get a Troubadour to help her attempt a non-combat Arcane check.
So the other options on the check to acquire don't matter, but what your skills refer to do matter.
I hope that helps.
Kyra can use the Acolyte, but then must roll 1d4 (unskilled) + 1d4 (Acolyte) for Arcane in stead of 1d6 for intelligence. I guess 5 is a better average than 3.5, so it would be useful.
I believe this part is correct. Kyra can chose to roll her unskilled Arcane, and get the acolyte to help. I am unaware of a requirement to use Intelligence because you lack the arcane skill: Can you share where you read that?
As far as I am aware, the only time where you cannot roll unskilled is for recharge checks, for spells and the like.
Even if your character doesn’t have any of the skills listed for a check, you can still attempt the check (unless you’re trying to recharge a card; see Recharge on page 15), but your die is a d4.
This has been clarified in the forums to mean you can always use a d4 for a skill you don't have, except for recharging a card. For this reason, they added the word "Even" to the beginning of the above sentence.
(Personally I think it would have been clearer to replace "any of the skills" with "a skill". This appears to be the intent.)