Dyes, their derivations and their reaction to air, light and washing are all terribly complicated and involved subjects.
At the present time there is little (read no) positive information that can be summoned to answer such questions.
This is one reason we have started the Weaving Art Museum's Research Institute -- to provide a center to begin working on the data that will be necessary to address these difficult issues.
Presently the funds are not available to actualize the plans we have and hopefully soon we will be financed enough to at least make a start.
The queries you post relate to our project and surely when results become available we will make them public.
Until then all we can tell you is there are no rules of thumb we could suggest to you to help you know what you are asking.
In that regards, we are reposting part of what you wrote here so we can specifically address a few of the points you raise:
"Does the predominate color of brown, mahogany or red—or their near cousins—have any significance in terms of age." No, the choice of colors is naturally related to the specific areas in which the materials used to make a weaving were dyed. Clearly, however, where the materials were dyed does not necessarily correlate with the area where the weaving was actually woven. This is a very important point to remember and one which is never addressed by other writers and researchers.
"I am aware that some fading and softening of the colors occur over time but I am under the impression that what is seen now is close to the original hue and intensity." Generally we agree but only when the weaving has not been subjected to a high degree of various of degenerative actions like air, light and washing.
“Regardless of where I look or ask I am unable to surmise and no one has been able to provide a useful and believable explanation of the significance of these color variations as the predominate field color for Turkmen weavings." Well, each of the various dyes components that will result in a 'red' color will yield different 'shades' depending upon the specific dye 'recipe', the different mineral compounds found in the local water supply and the various mordents used to 'fix' the dye into the material being used. Plus different types and grades of wool, silk or cotton (the usual materials found in the majority of Near Eastern weavings, Turkmen as well) will also play a significant part in the end results. Many factors and variables, so you can see how complicated these issues are.
"In most published data and in exhibitions there is at least an inference by what is exhibited that the older pieces were made in the brighter reds." To this we would have to remark we do not agree. However, older (i.e. what we call historic) pieces always have BETTER QUALITIES of dyeing. So if you equate better with brighter, then yes we agree.
"Do some reds that were once bright fad(sic) to a mahogany/brown? Ever, occasionally, never?" Great dyes never fade on their own, only when the degenerative influences we cite above have been at work."
Are there examples of mahogany/brown Turkmen weavings that would at least approach the age of examples that are catalogued as archaic? Yes!