Is there a way to hide the unread count badge in Mountain Lion's Mail.app?
With the advent of Notification Center, the display of badges on any app can now be managed alongside display of banners, popups, and other notification-related settings.

Go to the Notifications preference pane in System Preferences, and select the Mail app from the list - the option to turn off the Dock badge should be there.

You may need to restart Mail for this change to take effect, though on 10.8.4 I do not.