I just figured that if I had this problem, someone else might. I managed to fix it, but wanted to share what I did to do so.
There is apparently a conflict between the plugin wordpress-password-register and new-user-email-set-up.
What happens is that when used together two registration emails and two admin new user notifications are sent. After digging through the code for a while, I managed to figure out that it was caused by the fact that wordpress-password-register makes a new call to the function wp_new_user_notification.
So the fix is simple:
1. Find the new function wp_new_user_notification in wordpress-password-register and completely delete it. We want the function replaced by that of new-user-email-set-up. Don't actually replace it though.
wp_new_user_notification($userID, $this->user_pw, 'wp-pwd-register');
3. Comment out this line in your wp-login.php:
so that it reads:
What this does stops wordpress from trying to send out emails for new user registrations. Then it uses wordpress-password-register to take care of all that password junk, THEN send out a new user email notification. This allows the email that is sent out to contain the new password (instead of the one that is automatically generated) AND the message that you laid out in new-user-email-set-up.
Hope that helps anyone with the same issues. If anyone has a better suggestion as to how this should be done (especially if it doesn't involve commenting out a piece of wp-login.php), let me know. Because now, every time I update wordpress... I'm going to have to make that same change.