Why Tokyo police officers directing traffic wear face mask?

Answer 1

I think you're referring to the medical face mask. It's quite common for people (in the cities) in Japan and other Asian countries to wear face mask because they think it helps to reduce the pollutants they inhale in the streets. There is no stigma for wearing mask in these countries. If you do this in the US, you'd get stared at all the time.

At least in Japan, generally, people also expect you to wear a mask if you have a cold, even kids. They'd stare at you if you keep coughing without a mask. One time, a friend of mine was given a mask by a bystander because he kept coughing.