Activate Windows 8

Hi
To find your windows 8 edition, please follow:
Right-click the Start button and select System. (If you don't have a Start button, press Windows Key+X, then select System.) You'll see your edition of Windows 8, your version number (such as 8.1), and your system type (32-bit or 64-bit).

Also ensure, the product key you are used is not used to install on any other systems. Get the OEM product key purchased along with the laptop or get an unused product key.

Regards 

The first problem is that your screenshot is for Windows 10 Home and you are trying to install Windows 8 Pro.  Pro and Home won't share licenses.  You'll need to get a Windows 8 Home installer.

Secondly, try all three keys that are listed once you have the right installer.

you find the bios version if you go into bios with F1 or F11

Hi, as gentlemens above me said, you cant use pro version of licence with home licence. From my experience win 8 and win10 dont have big difference in system requirements and win8 will eventually go out of support the same thing that is happening to win7. So if there is no special reason to downgrade to win8 I would suggest you to stay on win10 if you have problems just make clean install or reset this pc.

The embedded key in the hardware is for the OEM supplied Win8 RTM Core
This is only officially available via a PC manfacturer

You can probably download the image for this from the OEM site using the serial number of the PC
https://support.hp.com/us-en/document/bph07143

Or (rarely) the machine is supplied with reinstallation media (on this model it would have been supplied as a .wim file in a hidden partition but the move to Win 10 would have removed this.

If you want to add a different version of Win 8 you'll need a valid activation key for it the embedded key won't be recognised which is what you are seeing.

Does it still have the HP recovery manager installed? If so you can run that it will restore the unit back to it's original state.
You can also run it when booting by pressing F11 repeatedly then choose keyboard type then choose troubleshoot then recovery manager then pick windows 8. 

i never found a key for 7*$ ...looks  a fake

Inexpensive activation keys very often are Volume License Keys that are being improperly and over used.  They may activate now, but may fail to do so later after Microsoft sees that they are being over used and shuts them off from future activation.