error in your FAQ

How can I styling the search form?

You can simply refer the awqsf-style.css.css that come with the folder and alter it or override it at your theme css file.
This file in bold doesn’t exist in your plugin.
The only .css file is set in ../wp-content/plugins/ajax-wp-query-search-filter/classes/scripts/default.css

Modifing or overriding this it works.

Have a nice time.

Angelo

http://wordpress.org/plugins/ajax-wp-query-search-filter/