Face value in note trading almost always means the unpaid principal balance, the amount of principal the borrower still owes today. It's not the original loan amount and it's not the total of the remaining payments. If a borrower originally borrowed $120,000 and has paid the balance down to $100,000, the note's face is $100,000 and a price of "80" means you'd pay $80,000.
The $20,000 isn't a windfall, and you had that right. Let me walk through where it goes. Suppose that $100,000 balance carries 7% interest with 20 years left, and the monthly payment is about $780. You paid $80,000 for the right to receive those payments. Over time you collect the interest plus the full $100,000 of principal, so the discount shows up as extra yield spread across the life of the loan. Your return on the $80,000 you actually spent works out higher than 7%, because you're earning 7% on a balance larger than what you paid. If the borrower refinances or sells next year and pays the loan off in full, you get the whole $100,000 at once and the discount arrives quickly instead of slowly. That's upside you don't control.
On your last question, yes, the borrower gets notified. When a mortgage loan changes hands, federal rules generally require notice to the borrower telling them who owns the loan and where to send payments, and your servicer handles that paperwork. A loan servicer is a licensed company that collects the payment, manages the escrow for taxes and insurance, and sends you the money, typically for a modest monthly fee per loan. You never have to speak to the borrower, and most buyers prefer it that way. Confirm the current notice requirements with your servicer in writing, since the details are specific and they do this daily.