Display one post on category.php: Wrong featured image

Do not use query_posts, and also do not replace the main query with a custom query. This causes more issues than solving the one issue you had before with the main query. I’ve done a complete post on these issues recently that you can check out here

According to your query, the only two things you need to change is posts_per_page and order to do this, you can simply use pre_get_posts to alter the query variables before the main query is executed. You can specifically target only category pages with the conditional tag is_category()

To solve this issue, return to the default loop on your category page. You’ll just need the following on category.php, nothing more.

<?php 
if (have_posts()) {
    while (have_posts()) {
        the_post(); 
        get_template_part( 'content', get_post_format() ); 
    }
}
?>

In your functions.php, use this code with pre_get_posts to alter the amount of posts per page and how they are ordered on category pages

function wpse_change_category( $query ) {
    if ( !is_admin() && $query->is_category() && $query->is_main_query() ) {
        $query->set( 'posts_per_page', '1' );
        $query->set( 'order','ASC' );
    }
}
add_action( 'pre_get_posts', 'wpse_change_category' );