Registration form Plugin… Email confirmation

I might be wrong but I think wp_insert_user function doesn’t call wp_new_user_notification function. As your edit you need to call this function manually.

As you are testing your plugin in a live site I presume the error reporting is turned off and hence you are not getting any error.

if (empty($errors)) {
        $myplugin_newuser = wp_insert_user(array(
            'user_login' => $user_login,
            'user_email' => $user_email,
            'nickname' => $user_nickname,
            'user_pass' => $user_pass,
            'first_name' => $user_firstname,
            'last_name' => $user_lastname,
            'description' => $user_bio,
            'user_url' => $user_url,
            'user_registered' => date('Y-m-d H:i:s'),
            'role' => 'subscriber'
        ));
}

if( !is_wp_error( $myplugin_newuser ) ){
    wp_new_user_notification( $user_login, $user_pass);
}
else{
    // this holds the errors that might be holding you back 
    // $myplugin_newuser->get_error_messages()
}

I think there might be some problem inserting user, or else your code should work fine. Just check if the returned value is a WP_error.