The thing with product key is that it depends. For my tablet, it was for sure not embedded into the BIOS. I re-install fresh off the Windows 10 ISO using Media Creation Tool and the key was gone. It could be a difference in version (OEM vs Home vs Pro), I can't remember for sure. But once I tracked down and install the original Windows image from the manufacturer, the product key is there.
Use 64bit version of Android X86 since HP is showing it as a Z3795 Baytrail CPU. Give Live mode a try and see how it works. You may run into black screen, lock up, long boot time, and screen orientation being off. It will be a long process to nail down all these problems. I went thru building at least a dozen custom kernel and ISO images to get to where I'm now.
Jokester: can you expand a bit more on your comment? In what ways do you think Android is clunky for HTPC?