When my son turned two we made his crib a toddler bed, and he would always get out. It was a nightmare since he couldn't understand the need to stay, but at 3 it should be easier for you. Once my son turned 3 we got him a bunk bed with a full mattress on the bottom, which is where he always sleeps. Our guy also likes to play for an hour, so our rule is, you can quietly play as long as you stay in your room. He does still bother us sometimes, but he always goes to bed on his own after a while. Let your son have his play time, as long as he stays in there and quiet, that will make the transition much easier. It will take some time to adjust, just make and stick with rules of when he can come out, what he can do in there, etc.
I'm guessing your son isn't potty trained if you want to lock the bathroom. You can do that, but realize that when you do potty train, you will have to leave it unlocked. We didn't have that problem since my son's bedroom and bathroom are both right beside the living room so we could hear him, (he was already potty trained by that time, so he had to have access). He also loves playing in the sink and getting in the tub, so I understand where you concern comes from.
For the actual bed, a queen is huge, but a wiggly kid could still fall off. I knew my son was ok on the crib mattress just above ground level, since he wouldn't fall out of that, so we just put him in his bed. Maybe you could put it on the floor as others suggested, and see how it goes. If he is fine, you shouldn't need a side rail.