HI,
1. When you edit a module via the menu Display>Content modules, you can select the target menu item of the links in the "menu" setting.
If you don't have the same menu item selected, you'll get a different menu item alias in the URLs.
2. Also, a simple solution is to use the "Force canonical URL on listings" option of the HikaShop configuration to auto fill the canonical URLs of the products so that the same link would be used on all the listings, regardless of the "menu" setting.
3. The automatic filling will be done only when the system needs to generate the URL on the frontend. So as long as you don't display the products listing on the frontend, you won't see the canonical URL filled in the product.
4. It's the relative path of the URL which is put in the canonical URL field, but it can also accept the full path.
5. There is no cache in HikaShop. So changes should be automatic. The only reason it wouldn't would be some other cache somewhere.
6. hikashop-menu-for-module-813 is the alias of your menu item. You can change it when you edit your menu item via the joomla menu manager.
"category" is the "category sef name" that you configured in the HikaShop configuration. So you can change it if you want.
28 is the id of the product, that you can remove if you turn on the "remove category and product ids from URLs".
original-buff is the alias of the product so you can change it in the product if you want.
/index.php and .html are added by joomla.