Keys are stored in the firmware on the motherboard. Shouldn't be an issue.

8.1 and newer devices have the key in firmware, and PCs upgraded from older versions of Windows are recognized automatically by the activation servers. So in most cases if a PC has ever had Windows 10 on it before, you can just install without a product key and it will activate itself.

Yes, it did on the desktop. I did buy a professional licence for the laptop though (it came with home).