How WP decide to show or not to show in admin panel the pop-up window with hint? Need a fix

Dismissed pointers are stored as user meta, you can inspect this for yourself with:

$meta = get_user_meta( wp_get_current_user() );
print_r( $meta['dismissed_wp_pointers'] );

In your case, the meta might be empty or damaged, to update dismissed pointers for ALL users on your blog, you could run this function (only once):

function wpse80084_dismiss_wp_pointers()
{
    $dismissed = array( 'wp330_toolbar','wp330_media_uploader','wp330_saving_widgets','wp340_choose_image_from_library' );
    $dismissed = implode( ',', $dismissed );
    $users = get_users();
    foreach ( $users as $user )
        update_user_meta( $user->ID, 'dismissed_wp_pointers', $dismissed );
}
add_action( 'admin_init', 'wpse80084_dismiss_wp_pointers' );