To me a nail pop is when there is a bulge outward, not a depression inward. It looks to me like a properly set screwhead and a poor job of patching. This did not occur because your ceiling is sagging.

Maybe sunk too far, maybe somebody put some weight on it from the attic. Take Joseph's advice.

If the drywall is not loose (push on it to see if it moves), the somebody simply forgot to mud the dimples. It happens. "Mud it, paint it, and forget it" is good advice.

I agree. If you push up, and the drywall does not move, fill it.