In D7 how can I append an element to the DOM via ajax on an arbitrary event (not forms and not links)?
First visit example.com/ajax/insert to verify your content is being printed out.

Your overall JS structure looks odd, plus you have a lot of typos. This .field-name-field- is probably not a real class since it ends in a trailing dash. /q= is not a HTTP query, you probably want /?q=.

Try this:

(function ($) {

  Drupal.behaviors.my_mod = {
    attach: function (context, settings) {

      $.ajax({
        type: 'GET',
        url: Drupal.settings.basePath + '?q=ajax/insert',
        dataType: 'html',
        success: function(response) {
          $('.field-my-field-class').html(response.data);
        }
      });

    }
  };

})(jQuery);