I'm answering the part in bold.
No. You do not want to do any training at 2:00 am. For a middle of the night potty trip, hark back to when he was just a wee guy, and would go out, potty, and you would carry him back in, and (hopefully) put him in his crate. This is what you need to do, minus the carrying. You DO need to put him on a leash or some other kind of lead for the middle of the night trip, which serves a dual purpose. One, you can see he actually went, and two, you can reel him in and take him back to his bed with no play, no interaction, no stopping in the kitchen for a treat. Hopefully, it won't last long enough to go into the colder months, but you're in Cali, so it shouldn't matter much.
Forgot to add:
He is not looking at you obstinately. Dogs don't have the same emotions that a human would. Please don't assign these to him. He's a dog, not a human. And too, he doesn't get revenge on you, and if he does something you consider bad, he's NOT showing guilt when he cowers, hides, etc. He's reacting to your body language.
Don't call only when it's time to stop playing. That teaches him that when mom calls, the fun ends. Call, and give a treat, a pet, a toy, or what have you, then release him to go back to playing. He will need distraction training, but not for this. That's what classes are for. Playtime is playtime. You make it a teaching moment by what I said above: Call, treat or praise, send back for more play. Oh and don't call if you think there is a chance he won't come. If he doesn't, then you are "poisoning" the call word.