Flour in the roux

According to On Food and Cooking by Harold McGee, roux functions as a thickener due the starches in the flour swelling up and interfering with the flow of water.

In fact, he indicates that technique can be used with any starch and any fat.

This implies that a lower-protein flour (which implies higher starch, weight per weight) like soft summer wheat, or a cake flour will make a more effective roux.

As an aside: roux can be made with oil instead of butter. As we know oil contains no water, and gluten only forms in the presence of water, we know that roux functions without gluten, as there can be no gluten development in an oil-based roux.

McGee explains that cooking the roux initially increases its thickening power by cross-connecting some of the starches. However, as browning occurs, the maillard reactions are transforming starches and proteins into other molecules, and reducing the ability of the roux to thicken.

So yes, it is true that the darker the roux, the less thickening power it has.

In New Orleans style gumbos, for example, the roux is so dark (almost a mahogony color) that it adds no thickening power at all to the stew—it is there for the flavor. The thickening in that dish comes from (depending on the tradition followed) either file powder or ochra, or both (not considered traditional).

I was not able to determine an ideal ratio of flour to fat. McGee indicates that a 1:1 ratio is traditional. However, if that fat is butter, about 20% of it is water, so that really does leave more flour than fat after the water evaporates. In any case, as long as there is sufficient fat for the roux to be cooked smoothly, it will work fine. Additionally, the fat in the roux is also indirectly an ingredient in the finished dish, so it might be desirable in its own right, depending on the outcome.