Shamim's Plugins

Can you please install a cron plugin (eg. WP Crontrol) and check if cron is firing correctly?

How many users do you have?
This plugin makes a notification request to query new messages every 2 minutes by default. You can change this by using following code

add_filter( 'fep_filter_ajax_notification_interval', function( $interval ){
    $interval = 5 * MINUTE_IN_SECONDS * 1000;
    return $interval;
});

After adding this code, it will make request every 5 minutes (you can change this to any number). Purge all cache and wait some time. Then check.

Those are the settings. Do you have any caching setup? If yes, purge all caching.
If your website is open in some browsers, this call will continue to make until they close the browser or navigate to another page.

You can send email to admin using following code. Change content as you like.

add_action( 'fep_action_message_after_sent', function( $message_id, $message, $new_message ){
	wp_mail('admin@example.com', 'New Message', 'New message waiting for approval' );
}, 10, 3);

Is the message sent to Admin or any user?

Plugin customization is not included in paid version. after that also i try to provide code if it needs small code snippet.
But to achieve your required functionality it not only needs template editing but backed code as well.

If you know php you can follow following steps
1. Edit view-message-content.php template file to show select field of users with search capability.
2. Add an admin ajax hook or backend function to handle selected user
3. Set that user as participant of that message using insert_participants method

Please contact using https://www.shamimsplugins.com/hire/

Did you already added CC field there (I see in your previous message)? If yes, how you added that?

If no, do you want to give user an option to add cc there? Or you want to add automatically? If automatically, what is there user id which you want to add as a CC for all messages? So that i can write some code and give that to you which will automatically add that user to all messages for you.