Is there any solution, to add a new item link to main menu(or to some module) and check if user is logged in. If not redirect them to login page/registration page?Thank you

If you set the menu access to registered and use the hack to the category_item.php or item.php that you will find here in the forum that will work.
pm me if you cannot find the post

community.getk2.org/forum/topics/restrict-k2-items-full-text-to-1?xg_source=activity
I think this will help