Menu for taxonomies and posts belongs to taxonomy

This is quite simple to implement. Get all taxonomies via get_terms() and get the related posts with a custom WP_Query and tax_query.

// get all terms
$terms = get_terms( 'docs_category', array(
    'hide_empty' => true,
    'parent' => 0,
) );

if( !empty( $terms ) && !is_wp_error( $terms ) ):

    foreach ( $terms as $term ):

        // fetch current term_id
        $current_term_id = $term->term_id;

        // get posts by term_id
        $args = array(
            'post_type' => 'docs',
            'post_status' => 'publish',
            'posts_per_page' => 3,
            'tax_query' => array(
                array(
                    'taxonomy' => 'docs_category',
                    'field' => 'term_id', // Or 'term_id' or 'name'
                    'terms' => $term->term_id, // A slug term
                    'operator' => 'IN',
                )
            )
        );

        $the_query = new WP_Query($args);

        if($the_query->have_posts()):

            while($the_query->have_posts()): the_post();

                // Do your stuff here...

            endwhile;
            wp_reset_postdata();
        endif;

    endforeach;

endif;

Edit: Sorry i skipped the topic with the posts without taxonomy. But you certaintly get a feel for how it works. You could create another WP_Query and an array of all available taxonomies and use a tax_query with an operator ‘NOT IN’.