Shamim's Plugins

Remove previous code and add following code in your child theme’s functions.php

add_action( 'init', function() {
    if ( class_exists( 'Fep_Email_Beautify' ) ) {
        remove_action( 'wp_loaded', array( Fep_Email_Beautify::init(), 'announcement_email_send' ), 12 );
        add_filter( 'fep_filter_before_announcement_email_send', array( Fep_Email_Beautify::init(), 'filter_before_email_send' ), 10, 2 );
    }
}, 12 );

This is a non standard solution. This may break in future.

You can use fep_filter_send_email_participants hook to add your admin user id. That will send email to that user also. You can conditionally add your user id for only first message.

Sorry for late reply. I had to setup for mobile debugging.
I have tested it with a chrome browser in android phone and working correctly.

Can you please deactivate all other plugins and change theme to default theme (eg. tweentysixteen) then test.

Currently you can use fep_filter_message_query_sql hook to change sql and return your desire messages.
We cannot pass multiple user id now to query. I have changed the query class so that we can pass multiple user id query. Next version you will get this. Then we will be able to easy achieve this.

You can see this changes in https://github.com/shamim2883/front-end-pm/commit/5786895fc0d01c07694c711331201942d2300b15

after this changes we can easily use hook like bellow

add_filter( 'fep_message_query_args', function( $args, $user_id ) {
	$args['participant_query'][] = array(
		'mgs_participant' => 4, //which user messages you want to show with current user
		'mgs_deleted'     => false,
	);
	return $args;
}, 10, 2 );

How many tabs are open of your website during testing?
How many users are testing simultaneously?
Please use fep_filter_ajax_notification_interval hook and increase the value to 300000 and test if that helps.

Let me know.

Currently archive message is shown in archive tab also in all other tabs as appropriate.

and thank you for suggestion. I will note it for future consideration.

I think you are not familiar with code.
Do you want me to correct this for you? If yes, please give me your cpanel and admin access. Please use https://www.shamimsplugins.com/sensitive-information/ to send credentials.