Could my cousin stay with me in military housing?

You need to provide a bit more information than you have. What is your rank? Why would your cousin need to live with you?

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.