My experience of entering the US with expired Green Card and NOA paper:
1. Standing in a lane for US citizens and residents, I moved fast to the immigration officer. He looked on my passport, expired Green Card, and NOA and send me to the passport office where the line moved very slowly :-( Finally, an officer checked my paper and some computer screen (I guess he accessed my status online) and then admitted me to the US.
2. The second officer told me when giving my NOA back to me: "Don't forget to send the paper (the NOA) back when you receive the new Green Card." I have never heard or read that people were supposed to send their NOA back to the USCIS when they received their 10-year Green Card. Am I ignorant or was the second officer?

The second officer was. My experience is that those officers know their own jobs very well, but their knowledge outside of the immediate domain of their own job can be very limited (if they were the sharpest students who graduated at the top of their class, would they really take that job?).

There's no rule that says to send the NOA back. There's no address to send it to.