Rabbits as a general rule don't like being held as it's too much like being scooped up by a predator. The peeing while you do it makes me think he was scared so he might be a floor loving bunny. Holding a rabbit on it's back so it "falls asleep" is called "Trancing" and I've heard conflicting...
Rabbits generally pick the spot they like as a bathroom. I would put a litter box in the spot he seems to prefer. The shoe box may also be too small a cat sized litter box works best in my experience.
He's growing up and full of bunny hormones. Honestly I've never seen a boy bunny "calm down"...
