If you want to find out what your windows activation key is, then you could simply use Speccy.
Install and run speccy, click Operating System and your key will be shown where it says Serial Number.
https://www.piriform.com/speccy
No clue why you care about a product code, although I`m sure you`re using the wrong term.
And as Jack has stated, the shop should have put a COA sticker on the PC or it should be on the packaging with the install disc.
Right there all over your picture it says, "This is not a COA"
But your activation key may very well be in your shot in post #6, it`s a 25 digit code in groups of 5 separated by a -
As stated, a Product Code is a barcode and not really important at all.
https://en.wikipedia.org/wiki/Universal_Product_Code
And a PRODUCE code is definitley not important
PLU-codes