Move my theme style.css into a folder

You can just register another stylesheet instead of style.css:

add_action( 'wp_enqueue_scripts', 'enqueue_theme_css' );

function enqueue_theme_css()
{
    wp_enqueue_style(
        'default',
        get_template_directory_uri() . '/css/default.css'
    );
}

This is, in my opinion, better than using style.css.