Do you realize that ducks and chickens cannot comprehend what playing is, and they don't do it as a result? Birds are too survival-oriented to waste time playing when they can be finding food and water. If you see birds fighting, they are never playing, they are actually being aggressive. So...
I agree. Ducks and chickens don't belong together anyway, and certainly not when one is a drake. If a drake mates a chicken hen, then he will kill her.
You cannot increase their egg production, that is something that you cannot control. Chickens begin laying more eggs when the weather gets warmer and the daylight increases, which is during spring and early summer, and then they decrease their production in the autumn and winter.