Hello,
I have just checked your site, and below is an answer for your questions:
<blockquote>(1) Only once the auto filter works. After the first action it can not filter next fields.</blockquote>
This issue occured because in your template, the div with id k2Container wrapped all body content, included left side bar (where JA K2 filter module is assigned) and right sidebar.
And when JA K2 filter get response from ajax request, it will update rusult for that div container, so your all content will be replaced.
To resolve this issue, please edit your template to update div id k2Container wrap only search result (item list) (you can check default template of K2 component)
<blockquote>(2) How many extra filed group can be to work with k2 filter & search ?</blockquote>
You can use many extra group in one instance of JA K2 filter module, as our demo here
http://joomla-extensions.demo.joomlart.com/index.php/ja-k2-filter/2-extra-fields-group