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
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.