I don't have kids, and when i separated from my husband, legal told me that i still have to support him. The amount i had to give him was almost 3/4 of what my BAH was, i think is depending on your rank. Legal said that if i pay the rent and other bills that total that amount of money than i don't have to give him anything else. So, kids or not, as long as you are legaly married, a soldier still has to support his dependent/s.
About the child support, that is totally different story, that is set buy the court not by the army, and she has to go to a civil court for that not military.
About losing the BAH if he is staying in the barrack, that is all up to his command. Where i am right now, in my unit, if you want to stay in the barracks, 1st- you have to have a good reason for it, and 2nd- you will lose BAH. Now, when i was in Campbell, we had soldiers living in the barracks and still draw BAH bc the commander alow it, so they can send the money to the family. So, i guess is up to the command if they want to let him live in the barrack or not.