Do you still qualify for BAH after getting divorced if you bought a house during marriage?

You will need to apply for an ETP for BAH from your post. Be aware that they probably will not grant it for more than a year, and you'll probably need to sell your house. The ETP is intended for people who are in a lease while they are getting divorced and can't break that lease but don't warrant BAH.

No. Buying a house was considered a personal expense and BAH is provided to pay for housing for individuals who are married or have kids or have attained a specific rank. You need to sell your house and will likely have to pay your mortgage with your base pay until it is sold. Is unfortunate, but I don't see many ETPs for divorcing couples who bought a house getting approved.

Doubtful, services say that your finances are your problems as long as you are paid correctly, if they do approve it will only last for maybe up to a year.