You wont need an air pump when using a sump as plenty of gas exchange will occur there. Thats what the black balls will help, they break up the water and give a large surface area for gas exchange, as well as providing a home for beneficial bacteria to live and break down fish waste.
The filter should be above the black balls. The black balls you don't disrupt nor clean. Leave them alone when servicing the filter. The foam, when above the balls, will trap particles. That foam can be cleaned or changed as needed.
Additionally, the water coming down from the tank should first go over the filter, down through the balls, then through a final sponge before the pump pumps back to the tank. I think your pump should be in the other side of where it is, the water level above the pump will fluctuate (it is there where you will notice evaporation), but the water level where the black balls and foam chamber is should be a constant level.
It is hard to really tell the exact configuration from your drawing. For example, the bio balls chamber usually will only have a gap on one side, yours also has holes on the opposing side which is not making sense to me. Also, how is water spilling over the top of the bio balls? It seems it is going through the holes in that tower straight through the bottom to the pump, its not passing through that media.
Here is an old photo of my sump, its changed a bit since I took the picture, but gives basic understanding.
Water comes in over the foam, through the balls, through the sump, then out to the pumps.
The black tower on the right is full of balls with foam layer on top. Additional foam block is behind the "Oceanic Sump" sticker near the center of the sump. That foam block area has a plastic retaining wall to keep the water level in the sump constant, while it spills over to the left side (left side of the sump is where you notice evaporation, notice the auto-top-off float there).
Hope this helps!