Answers:
Not exactly, the counts come from the app programming itself. Some have the count integrated, some don't.
You can actually change which apps are on the Dock (bottom bar) to whatever you like. And various apps have the new event count. For instance, I use The AT&T MyWireless app, and when my bill is due, I get a push notification there as well. Also when your apps have updates, you'll get one from the App Store icon. Any application that supports notifications will get a badge.