it is a good idea to be up on the rules of a cruise ship before departure. An adults only area means exactly that, no children allowed. No saving chairs means exactly that.
I have seen many "Parents" bring their teenage children to the adults only comedy act. These 13, 14, 15 year olds should not be there.
Many "parents bring their small children with them to the adult only pool and let them play there all day.
One last thing. Smokers are usually restricted to one side of the ship. If you don't want to smell the smoke please don't sit at the tables on that side.