I was sewing and I dropped the needle a few times. I was wearing boxers. I didn't feel anything any of the times I dropped it but when I dropped it on my foot later, the pain was very momentary and the needle bounced right off.

Could it have pierced my boxers and then my scrotum? Or would I have felt it?

Kind of a weird question. Do you have any puncture wounds? Are you suffering any pain? It's unlikely that this needle did anything to you under its own weight.

If you can feel yourself sit on your scrotum on silent you can definitely feel a needle. Turn of your lights, stand by the door and flash a flashlight around, the needle might reflect so you can find it, or run a magnet around

It is highly unlikely that a dropped sewing needle would cause an issue unless the angle of it hitting was right at the point.

Yes it can easily go though the boxers and into the skin.