Hi Tori,
To remove approval restrictions on new blog posts, you can head to the ACL section and allow the related user groups to publish blog posts(http://take.ms/4Xcql). If that permission is disabled, users under that user group cannot publish their new posts until moderators approve it.

As for the notification mails from another domain issue, we need to take a look at the site. Can you provide us backend and ftp access to that site so we can investigate this? Thanks.
If the domain is not licensed, you should add it into your licensed domain list to obtain support for that domain.