Does Mail have to be open for Notification Center to work in Mountain Lion? I upgraded to Mountain Lion and thought the Notification Center would be quite handy, however I've had a look around the web and it seems to me that emails only appear in notification center while the Mail app is open, or after it has been opened. Is this true, and is there no way around it? Surely the notification center must be able to check for new emails on its own, otherwise would the mail feature not be near enough useless?
The notifications that you receive for Mail within Notification Center are local notifications, not push notifications from some Apple server.  As such, the program has to be running in order for the action that will trigger a notification to be able to do so.

Notification Center cannot check for new mails by itself, it has no in-built capabilities for polling/checking etc in order to create notifications, it merely receives them and displays them.

It is possible that in due course some email app will be made available that supports push notifications, but at present none do, including Apple's own.  The fact that you get such notifications on iOS is more a feature that the mail client is always running in order to check for new mail, and create the local notifications accordingly.  Leaving mail open all the time on your Mac isn't really an issue, it's hardly resource intensive, and if upon receiving a mail alert you wish to act upon it, then opening the alert will only open mail for you anyway.

The best way to summarise it is that NC just listens for alerts that other applications create, and displays them according to your preferences.  It takes no part in the creation of alerts.