My cousin and I are fairly close. Would we still be able to hang out if he gets promoted to SGT and I’m a PV2?

What is AR 600-20 say about fraternization? Is your brother your 1st line supervisor? Is he in your unit? My former roommate, and close friend became an officer, of duty we are on first name basis. When in uniform we follow the protocol. That is the challenge, some people cannot turn off the "buddy" and dude during duty hours, that's why the Army is forced to create policies about fraternization.

Fraternization regs are primarily based on when a relationship is started and generally do not apply to family members. The military should keep you separate to where it never becomes an issue. You may have ‘barracks lawyers’ that try to tell you otherwise and the best way to overcome that is find the reg yourself and understand it.

Of course. As long as your cousin does not overstep any boundaries or you request certain favors. Fraternization rules are very easy to follow.