Sounds like both of you are making completely unreasonable demands on each other.
First, you. YOU WANT TO GO OUT EVERY NIGHT? Jesus.
Look, I may be, you know, one of the most extraverted people on this forum, but LET ME TELL YOU: That would be wayyyy too much 'going out'. In addition, when I have a goal that is extremely important for me, I have a one-track mind. Time stops. My mind is totally focused on reaching the target, and to hell with any idiot who tries to distract me.
This goal is so important to him that he's investing every bit of his strength into it. Why don't you just support the guy and say, "Hey, I'm here if you need me." It's better than trying to drag him kicking and screaming (figuratively speaking) into a world that doesn't fit his view. Honestly, if he's going out three times a month, he's probably making a HUGE EFFORT on his part to meet your needs. Ferk, I don't even go out to party three times a month. Hell, I probably party once or twice a year because I am extraverted in a VERY DIFFERENT way.
If you were to expect me to go out every day or even every week, after a while, I'd get very, very, very, very, very, very, veryyyyyyyyyyyyy grumpy.
Try to see it from his perspective. He's 'already failed' once. He cannot 'afford' to do it again. So he's investing all his energy into this. Support him.
On the other hand, this dude is also going unnaturally ape-shit crazy. Okay, so you slept with one of his best friends. Not cool. But if you weren't dating, that's just collateral damage right? No risk, no fun. How could you know you'd end up dating one day? He needs to calm his butt down a bit and take the oak tree out of his rear end. And if he has no evidence that you cheated (although it is rather suspicious) he's jumping to conclusions, probably because he's stressed to hell.
My suggestion: Let it rest. Wait until he's accepted into medical school - he'll be on a high, all happy with himself and in a super-great mood. THEN see what happens, and if things develop well, give it another shot.
But if you push it and push it, the worst case scenario is: He will fail again and blame you, even if it's not your fault. Then, you're screwed in the long-run with this guy because he'll always have that nagging him, and he'll always project his failure onto you.
So just be patient, and see what happens.