Misconception? "That gas is heavier so it sinks." I've heard people say many times that one gas is heavier than the other so the heavier gas sinks. Now this must certainly be true to a certain extent (I think of gas used in WWI that sunk into trenches). However, this idea of the weight of a gas being a driving force seems unusual because, in my experience, entropy is the dominating motive for particles in the gas phase. A good example of this is that gas phase clusters are studied extensively in the literature due to how unusual they are.

So, why would some gases "sink" and give up their entropic gain for what doesn't seem to be much of an energy minimization? Could the answer be simply that this concentration of heavier gases is quite short lived and they soon float away to their entropy filled wonderland?

I would also give as an example the fact that CO2 certainly gets up high in the atmosphere quite well and that is more than twice the "weight" of other particles in the air. So what's different about that than about the examples given previously?
Without significant mixing, diffusion takes a long time to mix gases.  Our understanding of entropy tells us that we will indeed finish with mixed layers, but that doesn't give us a time frame for that mixing, only an outcome.

Given a slow but steady production of a dense gas, layers absolutely will form due to density differences.  There are plenty of places where this can easily be seen, from your WWI example to the ability to pour carbon dioxide or hexane vapor into a container.  They're not stable over the long term, but they do exist for substantial periods of time, days in some cases.  

If confined by an open container, huge areas can experience this effect, as at Lake Nyos.  Heck, even small temperature based density differences drive atmospheric layering that has major impacts on weather and pollution.

It's not an enormous effect, but you may be underestimating the strength of the forces involved here.  The same pressure differential that forces a helium balloon upward acts on an open volume of helium as well, and more strongly due to the missing mass of the balloon.  A body of a gas like $\ce{NO2}$ has about as much force pushing it downward (in air) than an equal volume of helium has pushing it upward.  Without outside mixing to cause it to disperse, it will happily form a short term layer until diffusion carries it away.