Hi there. This is Steve, another Android expert here.
It sounds like you have the free 'ad-supported' version of Angry Birds. This version does require an internet connection to serve up the ads.
Can you confirm? Did you pay $0.99c for AB or is it the free version?
Let me know.
Can you give me the name of an app that you have paid for that is asking for an internet connection?
That is odd. The only thing I can think is that every so often the game needs to contact HQ to check updates, upload scores etc. It doesn't need to do this every time but every few days maybe?
You obviously don't have WiFi, head out to a McDonalds and use theirs, open all your games up and let them 'call home'. Maybe that will then allow them to be used without a connection for another period of time.