My advise was not a solution it was only for tests. As you can see when you disable SEF URL's all is working find which means one thing - mod_rewrite is not working properly on your server and should be disabled in Joomla! global configuration. Please contact with your hosting or check if the mod rewrite is even enabled on your server >> http://docs.joomla.org/How_to_check_if_mod_rewrite_is_enabled_on_your_server