Image and shortcode not showing in excerpt

To get shortcodes to work in the excerpt you’ll have to add a filter to your functions.php file (found inside your theme folder).

add_filter('the_excerpt', 'do_shortcode');

Your other question about the images not showing is hard to answer without knowing how your theme code is like. It has to do with how your designer build the category pages.

If you want to show the full posts instead of just the excerpt, find the file responsible to render the category loop (probably: category.php), and in that file where it says the_excerpt();replace that with the_content();