What would happen if a person had the right side of their body totally removed (including the brain)?
They would die.

Let's imagine that we perfectly cut a person down the middle. That would mean cutting the spinal cord in half, probably damaging the tissue so much that the remaining half wouldn't work well. The liver would be gone. The intestines and bladder would just be open. The heart would have to be adjusted to the left. The surgery would probably cause so much blood loss that they wouldn't even survive to die of other organ failure. 

And you already alluded to the mismatch between what side of the brain controls what side of the body, so if you kept the pseudocorpse "alive" on an IV, heart lung machine, and hemodialysis, they'd probably still be paralyzed.