Briefly reading the claims, it looks like only the apparatus and how it is constructed is claimed. The claims don't specify the rules of the game. For instance, it claims an elastic surface. If you make a version with a rigid surface you are probably safe from infringement. To infringe on a claim, you have to implement every part of the claim. You do, however need to recognize that there is potential for a different patent that claims rigid surfaces, or is not specific about the surface. Avoiding infringement of one patent doesn't guarantee you avoid infringement of all patents.
This is just an analysis based on a brief readings for the claims. You should also read the rest of the patent in case some aspects of the claims are interpreted in a nonstandard way.
With regards to your last question:
What would happen if the game was re-created under a different brand but with the same core idea?
The brand name doesn't matter with patents. If you infringe only one or more claims, you may be sued.