WordPress /page/2 not working

Don’t use query_posts, ever.

The first 6 queries can be condensed to a single WP_Query:

$args = array(
    'cat' => 3598,
    'posts_per_page' => 6
);

$featured = new WP_Query( $args );

if( $featured->have_posts() ){

    $featured->the_post();
    ?>
    your markup for the first post
    <?php

    $featured->the_post();
    ?>
    your markup for the second post
    <?php

    $featured->the_post();
    // etc..

    wp_reset_postdata();
}

For the main query, use the pre_get_posts action with a function in your theme’s functions.php. This avoids wasting a query when you overwrite it in the template.

function wpa_exclude_categories( $query ) {
    if ( $query->is_home() && $query->is_main_query() ) {
        $query->set( 'category__not_in', array( 683, 3598 ) );
    }
}
add_action( 'pre_get_posts', 'wpa_exclude_categories' );

You’ve now chopped 6 queries from every page load, and your pagination will magically work, do a little dance.