Could my cousin stay with me in military housing?

Third question first: If you are occupying government housing, you do not get BAH. First question: If you occupy base family housing, you may have guests in your quarters, no special permission needed. If you are in troop barracks, no. If you expect your cousin to be a member of your household, you need to establish dependency.

I'm going to assume a couple things so I can answer this, 1. that you're not married and 2. that you're coming in as a private.
With that, no, your cousin cannot stay with you on base, nor will you receive bah. You will live in the barracks, most likely with another private sharing a kitchenette and bathroom with individual bedrooms.

You need to give more information and the reason why.

Its possible depending on the circumstances.
When I was a young SST in germany my eldest brother and his wife were killed in a car wreck. I wound up being the one they were sent two. I lived in Quarters and had to have a Haus Frau live there also for when we went to the East-West German Border, Reforger, Gunnery, etc, etc. She saw them more than I did (but that was true of all families) and she was really upset when I came down on orders.
Either way some type of government investigators will ensure they really are your cousin. They did for my nephews and niece even though the civilian judge spoke with CID (FBI did the verification because of my clearance).

NO. Billets/barracks are for soldiers. You could get him a room at one of the visiting guest quarters on Post. Check in to it.