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Do the email addresses entered have to correspond with registered users of my site?

No! Anyone you choose to allow to be notified can receive an email. However, if you want them to moderate your comments that user must still have access to an account that has the permissions to do so.

If I use your plugin AND I have the Options->Discussion->E-mail me whenever check boxes checked, what happens?

Comments notifier was designed to work independently of the built-in comment notification functionality of WordPress. The Options->Discussion->E-mail me whenever check boxes have no effect on this plugin. If you have the above options checked, the email address listed under the Options->General->E-mail address field will also receive an email as is standard with WordPress. If you choose to leave those options unchecked, the main administrative email address will not be notified and Comments Notifier will still function and send out the appropriate emails to your list.