Yes, my DS7 has 'body in space' issues; he had 5 yrs of ot/pt and 2.5 yrs of vt; and more recently water therapy and neurofeedback. He had the oral, feeding, and speech issues and severe SPD too.
The part where your son's hands don't belong to him are 'body in space' as well as the hand-eye coordination, which you're seeing with puzzles. Late jumper is also an indicator. Dismissing any activity that requires both hands is another dead ringer. My DS7 avoided using both hands like the plague, but then he was born with torticollis (neck) and it was like he had a stroke with one side of the body. We've had to really force him to use the left side (which was the torticollis side).
Imo, it's all sounds related. It's probably due to a snag in neurological wiring. In our case, it was tied to severe uterine constraints during the pregnancy and DS7's torticollis. Yes, sensory processing and/or dyspraxia could be an issue too.
What would I do? I would seek out an OT eval asap if you haven't done so already. I would see a behavioral optometrist for a full perceptual eye exam and possible vision therapy asap; no, it's not too early either. I would also look into the Burdenko method for water therapy and/or use some of his techniques like making snow angels in the water.
I'd also work hard on doing anything using both sides of the body evenly and to enhance bilateral integration. This includes crawling through tunnels, swimming, balance beam work, making snow angels on land and water, etc. What you want to do is make sure the arms are crossing the midline and thus both hemispheres of the brain are being used. I'm still trying to do exercises with my DS7.
