Here to end the debates once and for all lol

His penis in Monsters is real, compare it to every time we've seen it and it exactly looks the same, he's just sporting a half chub in the shower scene, further proving it's real as most prosthetics would've had it limp and straight down. Him playing with it, tugging it, and even cupping his balls at one point shows it's real and they probably wouldn't have made him do too much touching if it was fake. His balls look the same every time he's shown it, and the pubes are groomed similarly.
For the cut/uncut debate. He's cut, but whatever doctor cut him was super conservative with the snipping and left some skin on the head. His twin is cut, and VERY CUT. I suspect Cooper was born with a lot of skin and the doctor just cut to uncover his head, which happens.