Weight vs. Mass of Clouds
There are different senses of weight used within branches of physics and engineering. In the strictest physical sense "weight", is given by W = mg. i.e. weight is the force on an object due to gravity alone.  Clouds are not "weightless" in that sense, since they are not far enough away from the Earth to escape its gravitational pull. 

The reason that clouds stay up in the sky is, as you say, because they are acted on by an equal and opposite force, namely the surrounding buoyancy of the air. Rain drops are formed by water molecules condensing around some sort of nucleus (a speck of dust, for example). This is triggered by the cooling of the cloud, or lifting to lower pressures, which reduces the moisture-carrying capacity of that particular "parcel" of air. 

While the arrangement of water molecules in the cloud make it light enough to be buoyant in the air, the arrangement of those same molecules in rain drops are not &ndash; so they begin to fall.

By analogy, I would think of the fact that a ship is buoyant, despite its huge mass, but if you 'condensed' the ship into a lump of steel with the same mass, it would obviously sink. It's weight remains the same however, no mass has been removed, and the value of g has not changed. So to answer your second question, no, phase changes do not affect weight. (Although strictly speaking, in quantum physics it can do &ndash; but the amount is negligible here)

Gases which are "lighter than air" will eventually settle in a portion of the atmosphere where the surrounding gases are the same density. They are not weightless, nor do they have a negative weight. The molecules are still experiencing a force due to the Earth's gravity.