Shamim's Plugins

You can use any domain you want. But email must forward to php script as mentioned in https://www.shamimsplugins.com/docs/front-end-pm-pro/getting-started-2/email-piping/ . So if your piping email is pm@privatemessagedomain.com then from this email forward email to mentioned php script.

add following code in your theme’s (child theme’s if any) functions.php

add_filter( 'fep_menu_buttons', function( $menu ){
	unset( $menu['newmessage'], $menu['message_box'], $menu['settings'], $menu['directory'] );
	return $menu;
}, 99 );

Yes.
Add following code in your theme’s (child theme’s if any) functions.php

add_filter( 'fep_get_option', function( $value, $option ){
    if( 'can-send-to-group' == $option ){
        if( in_array( 'supervisors', wp_get_current_user()->roles ) ){
            $value = true;
        } else {
            $value = false;
        }
    }
    return $value;
}, 10, 2 );

Change supervisors with your supervisors user’s role.

You should not need to change any php code to hide directory for non admin. What php you had changed? i can correct if anything wrong in the plugin for that feature. let me know.

Also what php code you had changed to hide directory? Please remove that code so that we can investigate what is wrong with directory.

Please use https://www.shamimsplugins.com/sensitive-information/

what is the nicename of that admin user the message is sending to? who is sender of that message?
please deactivate all plugins and change theme to default theme then try.
Let me know.

please go to Front End PM PRO > Settings > General > Show Directory