Change from all posts to specific categories post on main page?

Your syntax is wrong, change:

 query_posts( array(
                'post_type' => array(
                            'post',
                            $include_reviews,
                            $include_screenshots,
                            $include_videos
                        ),
                        'paged' => $paged )
                    );

to:

 query_posts( array(
        'tax_query' => array(
            array(
                'taxonomy' => 'category',
                'field' => 'slug',
                'terms' => array( $include_reviews,
                                $include_screenshots,
                                $include_videos )
            )),
        'paged' => $paged,
        'post_type' => 'post')
);