wp_nav_menu not appearing correctly on category page

Don’t call Nav Menus by menu_id. Rather, define an explicit theme_location for each place in your Theme template at which you want to include a Nav Menu, and then use that theme_location as the parameter passed to wp_nav_menu().

In this case, you’ve defined a header-menu Theme location, so change this:

<?php
 wp_nav_menu(
        array(
            'container_class' => 'menu-header',
            'theme_location' => '',
            'menu_id'         => 6,
            'depth' => 4
        )
    );
?>

…to this:

<?php
 wp_nav_menu(
        array(
            'container_class' => 'menu-header',
            'theme_location' => 'header-menu',
            'depth' => 4
        )
    );
?>

Also: your pasted category.php code omits get_header(). Please verify that you are calling get_header() in category.php.