The examples you showed are pretty different. Especially in the latter it is indeed dust being blown in the air by the machines, but the shapes are more like clouds. These require higher density and edge sharpness values compared to settings for mist.
So let's say you want more like low mist. Usually the default settings are good starters for this. Density 0.006 is perhaps a bit high, but good enough for positioning and tweaking your cloud. Eventually you probably need to lower it.
Lowering the cloud to ground level and decreasing depth is definitely the right way. Especially because the thinner the cloud the faster it will render, although that also depends on the density of course. These two values determine the amount of samples needed for rendering the most.
What works excellent is using a constant colour @ 1 as cloud density shader. Then lower the cloud to ground level and adjust depth to your liking.
From there you have 100% coverage (because of the constant colour) so it's very easy to determine the density and edge sharpness you're looking for.
Use a bright sphere or something to asses density of the layer.
After tweaking is done, plug the cloud fractal shader into the cloud node and get rid of the constant colour.
You can indeed move the cloud by changing X or Z values for position and keyframe those. Be aware that nowadays TG2 automatically creates an ease-in and ease-out effect at the beginning and start of the animation.
For a constant movement you can use a "get frame number" function which feeds into a displacement shader, then into a redirect on the axis you would like to move the cloud on. Then feed that into a warp shaders' warp input and put the cloud density fractal in the shader input of the warp fractal.
I think this is how it looks straight from my head/heart.
What it will do is displace/move the cloud with 1m/frame. Logically you can add a multiply scalar operator after the "get frame number" to speed up or slow down the movement.
I would never animate cloud seed by the way, because then with every frame the cloud coverage changes completely.
Unless you're interested in psychedelic stroboscopic animations

Hope this gives you some clues/ideas.
Cheers,
Martin