Help identifying differential 65 Barracuda V8

I believe that is a 7 1/4”, and there should be some stampings in the edge of the ring gear to identify the gear ratio. That being said, for all of your upcoming upgrades, you should strongly consider replacing the entire axle assembly with either an 8 3/4”, which would allow you to keep the small bolt pattern if you desire, or an 8 1/4” which will have the large bolt pattern. With the play present in the rear, regular driving will probably not harm it, but with your upgrades, I’m afraid that it will not last very long.