Remove or disable URL's with product/listing?

Hi,

Yes. Enter the URL you want in the canonical URL setting of your categories and products. That way, the search engines will use that URL and won't flag the other URLs of the same product/category as "duplicate content".

Hi,

When you have a module with a listing of products/categories, the links have to direct to a listing menu item. You can select which menu item to use in the "menu" setting of the module. If you don't select any menu item, it will try to stay on the current menu item but since your current menu item is a cart menu item, it has to add the product/listing part in the URL so that it knows that it is displaying a product/category and not a cart. So usually, what you want to do is to select your main shop listing menu item in that "menu" setting of your module. That way, instead of "myweb/carts/product/listing" you would get "myweb/shop" or something like that.
Also, if you have the "force canonical URLs on listings" setting activated in the HikaShop configuration and that you have a canonical URL entered for your products/categories (recommended for better SEO), then it will use that canonical URL in the module regardless of what you have in the "menu" setting of the module. This will avoid "duplicate content" issues where you have different links pointing to the same page.