I’d like to edit this search button. I would like it is on the right together with shopping cart and appears like simple search icon without place for typing. Where can I edit header? http://prntscr.com/k3brrv
You can edit header structure file depending on header type that you use, in your case, it is xstore/headers/center.php. Do all the changes in child theme files.
If you need help with customization you may contact WPKraken team.