Retraining a soldier in XCOM 2
Can't say for sure, as I've always played with the retroactive awc mod, but as far as I know, retraining does not affect the awc perk, i.e. if you had one, you keep it, if you had none, you don't get one.
No, just resets the perks.
Yes, you can unequip the weapon and it's available to other soldiers.
Stays where it is, PCS don't have any requirement.