Einstein,
You're mixing two seperate physical concepts, mass and weight, and appear to be using them as a single concept. Mass, basically, is the amount of "stuff" under observation - atoms, sub-atomic particles. That's a constant in a closed system assuming that there is no beta decay (for example). Weight, on the other hand, is the effect of gravity (or any other acceleration) on the mass. That's not a constant. Mass is a scalar quantity and weight is a vector quantity.
If you actually believe that the mass of a freely falling object is zero just stand under a brick plied on top of a scale and dropped together from the tenth floor of a building. What you'll discover is that when the scale strikes you and the brick thereafter strikes the now at rest scale that you'll have a headache and the scale will have registered the weight of the brick.
You might also discover simple Newtonian relativity: the acceleration vectors for both the brick and the scale were zero with respect to each other thus the weight, in that frame and only in that frame, was zero.
In your above example of an object falling in a vacuum your logical conclusion "
mass must be zero" is wrong.