Hello, @Artur,
1. Your issue has been solved please check back your site now after removing the browser cache. Actually, you have the categories on your site but none of them have been assigned to any of the products: https://postimg.cc/3dh8GqMw I just assigned the single category to a single product for testing and it is showing and working now: https://postimg.cc/c6kqx5sj https://postimg.cc/crtzsxkg so you have to assign your products to the products categories and then those categories will appear in search category dropdown.
2. For the Language Icon, Please copy and paste the below-given CSS code to XStore >> Theme Settings >> Theme Custom CSS >> Global CSS and check back your site after removing the browser cache.
.et_b_header-menu .menu li#menu-item-2137-ru.lang-item a {
padding-left: 10px !important;
}
Regards 8Themes Team.