You are making a major error in stating gases weigh nothing (at the Earth's surface).
Gases have a low density but it is certainly not zero. The weight of all the air above you presses down on the earth's surface with a pressure of just over 14 pounds per square inch.
If I remember my school science correctly, the molecular weight of any gas occupies 22.4 litres at the Earth's surface. That means that 22.4 litres of air weighs just over 28g.
Certainly, 12 tons of coal will produce 44 tons of emissions - the extra weight is the weight of all the oxygen from the air that got used up during the combustion.
Now your question about how do you actually weigh a gas.
Let me use an analogy first of all. For this I ask you to accept that water has a weight. Yes? Now imagine the problem of trying to weigh water while you are already submerged under water. A spring balance or bathroom scales cannot be used directly to weigh it in those circumstances.
In the same way, it is not easy to weigh air when you are already immersed in air. You would need to weigh it in a vacuum to use that method.
I think the easiest method of weighing a gas directly is to take a non-deformable container, pump all the air out of it to create a vacuum. Weigh it. Allow it to fill with the gas you are interested in weighing, and weigh it again.
Another way to demonstrate the weight of a gas is to compress it into a container (even liquefy it) and you will find that the more you pump in, the heavier it gets,