First of all, lovely art you've got there, Joe. I can look at it and immediately see how the process has helped you.
As to answering your questions,
Wwhich, in your mind, takes precedence between the three? Banding, Defragmentation, or Fidelity? I have a feeling you'd say it's on a case by case basis. My personal tendency is to prioritize banding-elimination.
Removal of banding gets a bit metaphysical at some point, where just by the process of stripping it out I often feel I am arriving at something more robust, but robust in what sense I can't really tell you, it's not making the subject clearer only, or adding detail (often it is removing detail). The way I would liken it is that zen quality of getting something down in painting with few colors and few brush strokes, it's not even exactly 'control' what I am describing here, more like chiselling something until it arrives at some quintessential level. Of course I can't explain this much better, suffice to say I've been working professionally, using most of my techniques (more on this later) to the point where they're second nature and they never interfere with getting things done to the best of my capacity, they only give an added self-sustaining enjoyment of getting things down to that quintessential level.
Defragmentation as you describe it I sometimes fight against consciously, adding more shades and blurs and dithers when the subject matter seems to desire it although that creates more softness and complication where a clear cut cluster could best suit. It isn't very often that this happens, though.
Fidelity, honestly for small-ish arcade game sprites, single pixels will happen, but aside from that I've never felt that I am sacrificing fidelity by banding removal and cluster clear up. The more you practice, the more it'll become second nature, there are wonderful solutions for most problems, and then there's that 'quintessence kick' when you find them.
2. Why does this work? What is it about the absence of orphan clusters that makes good pixel art?
I think it's flow, two (or more) married pixels always have a direction, as a brush stroke. An orphan pixel is a directionless dot, it has its uses but it doesn't suggest motion or connection between parts. When the contrast of the piece is such that you can both see the connections of clusters but also see them as related by lightness and shade, then you have a very harmonious, moving, sculpted entity.
3. AA was discussed earlier in the thread, as something that is less necessary. Wouldn't sharper screens make AA more necessary than before? Or did I misunderstand that point
No because AA relies on blurry crts to buffer a color in a color. The end result is that the blurry CRT art looks like painted art, convincingly, not that it looks like good, crisp pixel art. When you look at pixel art on a sharp screen, you're never going to hide the fact that the art is made of pixels as effectively as on a bleeding CRT monitor, so why try? Show pixel lines (esp, perfect lines), show connections, keep it crisp. Would a game of tetris look more beautiful if you ran it through a gaussian blur filter? Digital art gets to be super crisp, this is an asset, why treat it as a deficit?
4. After a year of practice, where do the single and double pixel stand in your mind? As I understand it this exercise is meant to strengthen the integrity of your clusters, so that the single pixel becomes a tool, not a crutch. But there were many places where I knew single pixel AA would be the best solution, I merely refrained. I'm sure you don't think single pixels should be abolished; where do you feel they're appropriate?
I don't adhere to a no-single pixel rule in my daily professional artwork. It's just not practical for very 1. low-res artwork that is also 2. very cartoony stuff which is what I'm employed to do at the moment. I do try to minimize single pixel usage unless it's highlights, dot eyes, that sort of thing. Aside from that, my point of view is largely unchanged from when I last wrote on this thread: banding removal/defragmentation of shapes is the bedrock of my technique and I think it's the only honest piece of craftsmanship that should be communicated and practiced for the learning pixel artist. The rest is just fluff that we created (for good or worse) in a community bent on classification and enumeration of 'techniques'. This is very common, most creative online communities create jargon. I do not return to 'selout', for example, as an effective and intersectional technique but I do think about banding and pixel shapes every day, in my work.
Tangentially: my recent interest is in bit restrictions for palettes, esp. 3 bits per channel, like the Atari ST. I use this restriction in my daily work because it explains the color choices of past videogame artwork very effectively, a lot of linear from-hue-to-different hue ramps, full saturation, less use of gray, less use of interchangeable colors, a stronger foundation in pure black for segmentation of forms and so on. This is a blind spot in how we look at art in Pixelation, because we jumped into learning pixels with a 24bit palette in our hands and no matter how restricted and small the palette we used was, it was still picked from a color space of millions of colors. So many close shades, so many earthly greys, so on. It is a joy for me to un-learn, for a time at least, my dependence on shifting colors through grey buffers to get things done, and it's really no way to practice this with the c64 palette or the EGA palette, honestly. I encourage anyone with the means to restrict their bits-per-channel to give this a good go, especially with 3,3,3 and 4,4,4.