Check your mailer settings. The registration email address within CB > Configuration > Registration should be on the same domain your emails are sent from or left blank so it uses whatever you set the from address to within Joomla global configuration.

Joomla does not error if SMTP is not configured correctly. It will fallback to phpmailer or sendmail. So Joomla mass mailer won't indicate it's working. CB will error that the SMTP connection failed if credentials are wrong otherwise it'll provide a mailer error for whatever failed. Change to phpmailer or sendmail and see if either of those work.

Moderator notifications also need to be enabled within CB > Configuration > Moderation and your user within CB > User Management needs to be configured to receive moderator emails.