T.,
You mention that he is "still" peeing in the wrong places and when he has uses the common bathroom he chooses to aim for the garbage can. It sounds like he's been doing this for a very long time. At ten years old, I'm pretty certain he knows right from wrong in this situation. He has probably continued to do this because there were no consequences to his actions. I think it makes sense to have him clean up after himself.
Although finding out why he is doing what he's doing could help both of you, I don't believe in making excuses for poor behavior. Get counseling if you feel it's necessary and begin to deal with his actions today, lovingly. Don't ridicule him (I'm sure you don't), but just let him know that he needs to clean up after himself. And, hopefully, he sees that the other kids also have to clean up after themselves whenever they make a mess, whatever kind of mess it is.
He might just need a little more attention from you. He might notice that you give him lots of attention when you discover what he's done. Try to give him good, positive attention, even if you have to put something else aside.
As far as what to use to clean up the mess, I have found a cleaner at the pet supply (that works for people messes too) that is called Simple Solution. It removes the smell of urine and feces to the point that even animals aren't supposed to smell it. It also removes stains very well. I've used it on kid messes in my own home with great results.
I wish you the best.
God bless,
J.