Hi,
The issue with the category not saved in the badge comes from a bug. Download the latest version of HikaShop and install it on your website and it will add the patch that we added recently to correct that.
There is no point in having both a product selected and a category selected in your badge.
Just select the product if you want the badge to apply only to that product.
Also, you set 10 in the quantity field of the badge, so the badge will only display if the quantity of that product is exactly 10.