Can admin receive email notifications of new user registrations? I would've thought this would be core functionality, but does it require additional plugin functionality?
I believe you would need a plugin for this — but the plugin can be really basic. In your plugin's main file simply add an 'init' method that listens to the 'users.saverUser' event and checks the 'isNewUser' event attribute; then prepare and send your email.

The process is outlined in this answer that sends out an email to members of a userGroup when an entry is saved. Same principle would apply, just using users.saveUser instead.

public function init()
{
    parent::init();

    craft()-&gt;on('users.saveUser', function(Event $event)
    {
        $user = $event-&gt;params['user'];
        $isNewUser = $event-&gt;params['isNewUser'];

        if ($isNewUser) {
            // Send Mail
            $email = new EmailModel();
            $email-&gt;toEmail = "user@email.com;
            $email-&gt;subject = 'New User Registration';
            $email-&gt;body    = 'New User: '.$user-&gt;firstName.' '.$user-&gt;lastName;

            craft()-&gt;email-&gt;sendEmail($email);
        }

    });
}