Windows 7: HP notebook came as a gift, but besieged with malware.

Yes it does.

Which, if your gut feeling is right, it's a "fugazi".

Too bad for the girl. She's a nice kid.

Have postponed this project for a day (paying work came in). I will post back on the validity of this.

Not a gut feeling. If "OEM" is in the Product ID string then it is either factory or builder's OEM activated, or an illegal cracktivation.

It can't be Factory OEM if it isn't the licensed OS on the COA sticker.

It could be Builder's OEM if the uncle bought that instead of Upgrade version. I would ask him. Don't tell him why you're asking. He should at least remember whether he bought Upgrade or OEM version. If he insists its Upgrade then it likely means he or whoever installed cracked it.

A cracktivation is likely done at the BIOS level using a factory OEM key to emulate SLP factory activation like it comes from factory. I would wipe the HD first with Clean Command before any install: How to Clean or Clean All a Disk with the Diskpart Command

I performed the slmgr routine, and the key was rejected. AGAIN, nice uncle.
Good call on your part. And, I've been made aware about keys & Product ID's not jiving.
It is never too late to learn.

Called the gal up; explained everything; she agreed to buy a Home Premium key.
Will set it all up tomorrow.
Case closed.

All's well that ends well!

Same basic steps to get a perfect clean reinstall after wiping the HD with Diskpart Clean Command:

Clean Reinstall Windows 7 Factory OEM