I have found and solved this problem. Just an type error in CSS 

My fix was following. I discovered that there is a filed in config.php file that need to have same value as fields in CSS. Otherwise the font size will 'jump' up and down.

So if you for example want 11px font size in chat window, be sure to in config.php change:

  // Text size of the text in the chat window (in pixels)
        $conf['text_size'] = '11';

Now open your templates chat.css and live.css and change body, td, a, .operator, .guest font size to 11px.

That was the only way I succeed to achieve positive result.