Custom Queries – Child Categories not showing up

category__and is exclusive, so it will not work for your application, if you use tax_query and setup your query as follows, it should work:

$args = array(
    'tax_query' => array(
        'relation' => 'AND',
        array(
            'taxonomy' => 'category',
            'field'    => 'id',
            'terms'    => $city,
            //'include_children' is true by default
        ),
        array(
            'taxonomy' => 'category', //may need to be categories, I don't recall, tweak it till it works.
            'field'    => 'id',
            'terms'    => $category,
            //'include_children', again, is true by default
        )
    )
);
$posts = query_posts( $args );

If that doesn’t sit well with query_posts() (I don’t see any reason it wouldn’t), WP_Query won’t complain, so just use that. Also, that link to WP_Query has all the docs you’ll need to modify that query, so thumb through there, you’ll probably at least want 'post_status' => 'publish'…maybe some more.