What do my children do when I tell them to do their chores?

They will wish the floor. Child labour!

"Go do your chores, (Name)." — You can ask your child to do chores; they will begin to sweep the floor unwillingly.

I experienced back talking from some children, this just happens. In the case of back talk, they won't do anything.

Knocking things is, as @Ben mentioned, a poor collision detection.

Source:

http://www.uesp.net/wiki/Skyrim:Adoption

When you tell them to do chores, and al is activated which allows them to sweep the floors until you tell them to do something else such as go to bed or go play outside. running over stuff is just a collision problem.