How much does air weigh when weighed in air?

Answer 1

Yes, it does. The air has air pressure all around you, about the weight of 2 elephants. Then you would ask, Why are we not we crushed? It is because You breathe in that air and you have the same air inside you that is outside.
Yes.
Air does have a weight, yes.

Answer 2

Depends on the density of the air being weighed relative to the air around the scale. Cold/dense air weighs more than ambient air as cold/dense air has more atoms per volume.