Can you build a bunker under an existing house?

Depends on the building and safety rules where you live. 

Probably not. Apart from concrete foundations, there are electrical, water and sewer lines running underneath. 

You can in theory, but depends on local codes, and usually not if it already has a basement.

You could. But you would have to get the proper permits from the local government prior to building one.

Yes, you can do that.

Sure you could, it's just way more difficult.

Providing it meets local building codes and you get the proper permits. 

In the 1950s, many people built bomb shelters on their properties. We were convinced that the Russians were going to bomb us at any minute. Look up the plans for those. 

No, I can't. I'd have to hire someone to do it.