Since you're only concerned about capturing historical results, I would take out the tables and columns describing game rules and other meta-information. That will simplify your model greatly.

I'd also suggest starting off by modeling types of games in which balls are only drawn from one pool. In this case, for each game, you'll have a "game event" (a date on which a drawing is done) during which a number of balls are drawn.

If you're willing to take a crack at this and post an updated model, I'll help you work towards a final model.