iPhone push notifications?

No, the "default" apps on the bottom row aren't the only apps that can receive push notifications or display number badges.

There isn't necessarily a relationship between push notifications and number badges. Apps can have number badges without push notifications (Mint comes to mind) and can have push notifications without number badges (MLB At Bat, for example).

Any app that has push notifications implemented can have the number badge on the icon. Messages is another Apple app with push notifications. Some 3rd party apps also have push notifications.

Not exactly, the counts come from the app programming itself. Some have the count integrated, some don't.