Hello,
1) If the search by products is enabled in the Theme Options > Header builder > Search then our theme uses the shop archive page as the search results page no matter if products found or not, that’s why shop appears in the breadcrumbs. You can hide it using custom CSS below
.woocommerce-breadcrumb a[href="http://test2.scienedu.tk/?post_type=product"], .woocommerce-breadcrumb a[href="http://test2.scienedu.tk/shop"] {
display: none;
}
2) Fixed. Check the Product search widget now.
3) This is the default WordPress widget and we can do nothing with this. You can check the workaround mentioned here https://www.goldenoakwebdesign.com/blog/how-to-exclude-woocommerce-products-from-the-search-results-in-wordpress if it suits your need.
Regards