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Found a post on the XCOM subreddit from one of the Long War devs, Amineri:

I should clarify how this works.

The AWC doesn't give a random chance to gain an ability on each level-up.

What happens is that each soldier gets a single AWC ability rolled for either when the AWC is built (for existing soldiers) or when the soldier is obtained (AWC already built).

The AWC ability is randomly pulled from "cross class abilities" -- not a list, but configured with a boolean in each ability.

The AWC is stored along with the level at which it will unlock. When the soldier reaches that level (or returning from first mission if already there), then the ability will be unlocked.

Source: I looked at the code, and the LW Officer mod stores the officer abilities in the same AWCAbilities array.

So, from this, it seems the answers are, 1, a random selection from "allowed cross class abilities", and yes, but see this question.

Yes, you can disable the tutorial by unchecking the tutorial box on the start game screen. It's in the box of settings below the difficulty selection.