Emails and Messages are two separate and distinct things, as described below...
EMAILS come to your OPP webmail address, which you chose when you signed up for your paid site, and are sent by people from their own email addresses. You must log into the Webmail system to view your emails using the 'Send/Receive email using OPP Webmail' link in your Control Panel. You can also check your email with a POP3 program like Thunderbird, Outlook, or Mac Mail. Instructions on how to set up POP3 email can be found in the following POP3 Email.
MESSAGES are sent via your 'Contact' page on your website, and can be seen in your Control Panel. When you are sent a message through your site, you'll also receive an email notifying you that you have a message to be viewed, but the actual content of the message isn't viewable in the email.
A notice that you have received a message will be sent to both your OPP email and your billing email address. Then you can log into your Control Panel and click the 'You have X messages' link and check your message. The X will be the number of messages you have.
You will not be notified at your billing email that you have emails, only messages. Also, the number of messages noted in the Control Panel is not reflective of or related to the number of emails you may or may not have. You must log into Webmail or your POP3 email account to find out if you have any emails.