does the serial number which is etched in small type on the bottom of your MacBook Pro match the serial number which is reported by About This Mac on the Apple menu? If your logic board had been replaced by Apple or by an Apple-authorized service provider, then they should have set the replacement logic board’s serial number to match the serial number of your original logic board. If they didn’t do that, then you’ll need to contact whoever did the replacement to get that resolved.
Do you still have the original proof of purchase of your MacBook Pro? If you do, and your AppleCare Protection Plan was registered with Apple to your MacBook Pro before its manufacturer’s warranty expired, then the AppleCare should indeed last until the third anniversary of your MacBook Pro’s purchase. If Apple’s records don’t match your original proof of purchase, then you’ll need to contact Apple directly to get that corrected.
I have no experience with the Angry Birds app, so perhaps someone who does have experience with it will assist you with that.