No email notifications when Mail.app isn't running?
I can only say that this is true. Mail.app has to be open to get notifications.


  A workaround: Open Mail, and close it with the red dot (or command+w). So the blue
  point under the icon in the Dock is still there. So you have no window
  open but mail is still getting emails.


There is no app or anything there yet to fix this "problem".