iPhone push notifications? I'm fairly new to the iPhone, so apologies if this is an obvious question.

Are the default programs on the bottom bar (Phone, Mail, Safari + iPod) the only applications that can show 'new' event counts? Is it to do with them being on the bottom bar?
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).