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.
If your soldier gets gravely wounded in a mission, there is a chance that the soldier gets shaken. This chance is 20/25/25/30% for the four difficulties.
What is gravely wounded? It means that the recovery time is more than 168 Hours (336 on Legendary) - this is determined at the end of the mission.
How is recovery time determined? Based on the lowest HP percentage at any point in the mission, a number is generated within certain limits. There are four "buckets": up to 20%, 21-50%, 51-75%, 76+%.
Example, your Soldier has 10 HP and takes 6 Damage, which brings him down to 40% health. At the end of the mission, the game looks in the 21-50% bucket to see the range of how long recovery takes and picks a random number from that. If that random number is more than 168 hours (336 on legendary), you're gravely wounded and the game rolls if you're shaken.
Note that the source of damage doesn't play into it, and neither does healing. The game records the lowest your HP ever was. E.g., 10 health, take 6 damage, heal 4 damage, take 3 more damage, heal 4 damage -> Even though you end the mission with 9 HP, the game records that you were down to 4 HP, which is 40% of 10. That percentage is the ONLY thing that counts.
I do note that the spread between best and worst recover time is MASSIVE. For example, a single point of damage (the 76%+ bucket) on Rookie difficulty puts your Soldier out of commission for 5 to 25 days - completely random.
For more technical details, I dug into the source code here.
Bonus Answer: If you recover from being shaken, you get a permanent increase to Will (4 to 13 points) and can't be shaken again - it's like Vaccination.
Best Answer
Have the burning member run through a tile that contains water. This will put out your soldier and remove the Burning debuff.
This can include tiles such as Fountains on certain tilesets or puddles, even the green goo encountered in the first Blacksite mission you go through. If the weather is raining it will not put out the fire.
Medkits will also remove any status debuff, like Burning or Poisoned.