The AppThemes demo site is logged in with an admin account by default, so whilst you can see an "edit post" link on a listing on that, a regular user account would not see this on the listings. Customer accounts can edit only their own ad listings and this is by way of a front-end editing page, they do not access the WordPress admin dashboard.
It would require customization on your part if you wanted to add other search fields to the main index page, however you may wish to note that ClassiPress includes an advanced search functionality which is visible on the category view page as shown here on the
ClassiPress demo site. You as the site admin can determine which fields appear in this search. Thanks.