If you find this interesting, I suggest you study group theory - this seems pretty much a direct consequence of the group structure.

Not as far as I can tell from skimming https://en.wikipedia.org/wiki/Random_permutation_statistics.

No.

The exposition has its problems too. Consider:

>> Zero fixed points — not a single hexagram occupies the same position in both orderings. The structural difference is total.

As a mathematical matter, the expected number of fixed points for any permutation is 1. Some have more. For some to have more, others must have less, and all of those will have 0.

But as a logical matter, "the structural difference is total" is pure gibberish. Consider these two permutations on 5 elements:

    1. [2, 3, 4, 5, 1]
    2. [5, 1, 2, 3, 4]

But of course these two permutations have a nearly identical structure (they are rotations in opposite directions, and are each other's inverses); they are far more closely related to each other than either is to

    3. [4, 3, 2, 1, 5]

Then:

>> We reframe the question:

>> Transform the question "what is the structural distance between two orderings"

>> into the mathematical problem "what is the cycle structure of a specific permutation in S₆₄?"

This is nonsense. The 'question' cannot be transformed into the 'problem', because they are completely unrelated ideas. It's like transforming the question 'what is the Levenshtein distance between two strings?' into the problem 'if a specific string were in alphabetical order, how would it be pronounced?'.