#1. You will have to overwrite the archive-project.php file in the child-theme under the path pakrco-child/template-parts/archive-project.php then edit this file again. Use function get_terms to browse all project categories
#2. That code is fixed for the page project archive, it is not written into a shortcode or a widget for use in many other locations. So if you want to customize, you will have to rewrite this function.