As for popping the balls, could you use one piece of plywood, under the play field, with a pin for each hole.
Then have a solenoid move the plywood up and down. All the pins would move but would that matter?
The solenoid would have to be pretty hefty, requiring a power transistor to turn it on, and may be noisy, so this may not be desirable.
Or something like this, but you would need one for each hole:
http://www.allelectronics.com/make-a-store/item/sol-102/miniature-12vdc-push-type-solenoid/1.htmlAnd with just about any solenoid, you are going to need some type of driving circuit.
Edit: just read the customer comments on that solenoid, it may not work for what you want. I'd order just one first to try if it were me, if I decided to try them.
Edit again: If you used the one piece of plywood idea to move all the pins at once, you might could use a servo motor with a crank in the center of the plywood to move the plywood up and down in one servo revolution.
The panel wouldnt have to be plywood, it could be a lighter plastic sheet.