Hi Nikos,
Thanks for the info but pls don't put the info into your reply as it's in public, you should edit Fields in your first post and put info there.
As I can see that you already have JA K2 filter and search module created but not assigned to any position yet. You can follow this documentation to configure module:
https://www.joomlart.com/documentation/joomla-component/ja-k2-filter#module-config
In the module settings, if you don't want to show the search button , you can enable the 'Auto filter' option.
And you will need to have a menu item on which the search results will display.
About the Additional info on our demo site, that section will show the Extra Fields of each item. You can manage these extra fields in Backend >> Components >> K2 >> Extra fields:
I hope this helps.