Most of the rest has been addressed by Eric B (thanks, Eric!), but...
I think you mean Ni Fe Si Te Fi Ne Ti Se ?? Do you have any links on this Thomson's ordering?
That was what I meant... a bad point to typo. I apologize.
In the book, it seems to be the one listed above. It was apparently called the "lasagna model" because she basically took the "shadows" (the "other four", usually left out of function discussions) and placed them
between the dom/aux and tert/inferior. You can see this described here:
What's New In Type and Temperament
The other main source for information on the book is the "Exegesis Wiki":
The Lenore Thomson Exegesis Wiki
More recently, she seems to accept Beebe's model, though she doesn't go into a hard "order" either way. I assumed Beebe's roles matched hers (e.g. "Crows Nest"=Trickster/demon), though she didn't exactly say that. She does seeem to say that degradation into either set of archetypes is something that occurs during "ego-disintegration" and "individuation". Beebe and his followers seem to hold these things as something that can come up in daily stress, and that seem to make more sense to my experience. If I understand correctly, ego-disintegration would be more dire instances of stress, and of course, in Beebe's theory, they will come up then too. Individuation I do not understand yet, but it seems to be the goal connected with what
we would call "developing the functions", though she says it is not really about "developing" functions.
I think Thompson does make a distinction between normal, day-to-day, relatively conscious use and development of the functions vs their role in individuation. I think in her view, our egos tend to make up their minds (to so speak) and keep heading in the same direction as long as possible. Eventually, our previously adaptive behaviors and definitions of ourselves are not sufficient, and we need to change something.
The ego doesn't want to change even when it's current approach and toolbox of functional use isn't sufficient, and that's where our unconscious tends to grab the reins, and express itself archetypically (or at least against our conscious aims). At that point, our current conception of who we are needs to adjust or expand.
I feel that Jung's description of the process of individuation is nebulous and confusing. (By the way, if there was anyone who needed a good editor to force conciseness and clarity, it was Jung.) There are different aspects of it, but the basic concept is that we start as mostly unconscious. Then we embrace some aspects of ourselves, meaning we reject others (these become things like the Shadow, our Anima/Animus, etc). Over time we become aware of the rejected parts, and are forced to acknowledge them. Eventually, we learn how to claim them as parts of ourselves, even if they never become equally favored and preferred.
So, in general, the process of individuation involves choosing initially and making conscious (we are not usually aware of this because mostly it happens during childhood), developing the initial choice, become aware of the rejected, and then accepting the rejected in some way (including by expanding our definition of who we are, or seeing the rejected parts as part of a greater whole or cycle).
For Jung, this process happens with the Shadow, the Anima, the archetypal spirit (wise old man or earth mother) and finally the Self. Note that the Self here doesn't mean the individual self or ego, it means more a transformative unifying view/process that stands apart from the ego and moves toward true wholeness.
Individuation isn't a one time thing, but an ongoing process that happens over and over again in different parts of the psyche.
So one can see how a similar process could happen with the 8 functions. We make our initial choices, develop them, become aware of their opposites, and learn how to deal with those opposites. At first, the opposites of our primary functions are entirely the threatening Other, but eventually we see acknowledge them as parts of ourselves (however under-developed), and grant them some role in our lives (however minimal).
So I basically hold Beebe's order, since it is parallel. (The four shadows are the functions in the same order as the [primaries, but with the attitudes reversed). The brain lateralization theory would simply introduce additional "tandems" of functions. So the dominant would degrade directly into the "demon" or 8th function, and the aux. into the 7th.
I do think Beebe's order does better represent the order in which individuals of given types become aware (on some level) of the functions. As an Fi-dom, my early encounters with Te (and demands that I use Si) were intrusive, jarring, and definitely the Other. My awareness of Ti and Se (for example) came later, because they tend to blend more (internally, anyway) with my most preferred functions.
I do think Thompson's order does better represent a typical order of preference for functional use and development, though. For most INFPs, for example, their Te is not going to be their fourth best/preferred/most-efficient function, even though we can and do use it consciously. There are going to be exceptions to that

hi: thehigher) and it certainly isn't set in stone. Any number of environmental and personal factors seem to make a difference... otherwise there wouldn't be anything to debate on this and we would all agree.
I do like the "tandems" concept, as well, because my Ti tends to subtly back my Fi a lot, and Se steps in and expands what I can take in via Ne on occasion. It took me a long time to become aware of that, since it's a subtle process and it doesn't feel like a big clashing or shifting of gears. I also realize that in others (and between others) Ti and Fi, for example, aren't necessarily best buds.
Where I'm still struggling with Beebe is assigning specific archetypes to specific functions. I'm fine with his primary few, but the shadow functions feel arbitrary and don't fit with my personal experience. I've tried warping my brain around to try to MAKE them fit with my personal experience, but it just doesn't work for me. I've posted on it before, but my guess is that the nature of the operation of the function itself and one's early encounters with it are going to have an effect on its archetypal association and expression.