Deploy Subcategories with Ajax not working

onclick="cat_ajax_get(...)"

looks for a function in the global Javascript scope. This line:

jQuery(document).ready(function($){

hides everything from the global scope. So, this won’t work.

You will have to use a completely different technique here. Do not set the onclick attribute. Instead attach the handler from inside your Javascript code:

jQuery(document).ready(function($){
    var links = $(".filter a.ajax");
    var ajaxurl="<?php echo admin_url( "admin-ajax.php" ); ?>";

    function cat_ajax_get (event) {
        var catIDText = $(event.target).parent().prop('id');
        var catID = catIDText.substring(4);

        links.removeClass("selected"); // remove class from all lines
        $(event.target).addClass("selected"); // only add it back on the selected

        $.ajax({
             type: 'POST',
             url: ajaxurl,
             data: {"action": "load-filter", cat: catID },
             success: function(response) {
                 $("#category-post-content").html(response);
                 return false;
            }
        });
    }
    links.click(cat_ajax_get);
});