This topic is fascinating, but the two specific questions asked are trivial, as indicated in the [comment][1] of Peter Taylor. Here is a paraphrase of that comment. The function $W$ has the form $W(a, b) = w(a^{-1} b)$, where $w(g) = W(1, g)$. Let $f(g) = w(g) - w(g^{-1})$. Then $a$ is better than $b$ if and only if $f(a^{-1}b) > 0$. Let $k$ be the number of $g \in S_n$ such that $f(g) > 0$. Then for every $a$ there are exactly $k$ elements $b$ such that $f(a^{-1}b) > 0$, i.e., $a$ is better than $b$, and there are exactly $k$ elements $b$ such that $f(b^{-1} a) > 0$, i.e., $b$ is better than $a$. Moreover $k \ne 0$ because otherwise no element is better than any other, which we have [seen][2] is not the case.


  [1]: https://mathoverflow.net/questions/480917/generalization-of-a-mind-boggling-box-opening-puzzle#comment1252192_480917
  [2]: https://mathoverflow.net/questions/480917/generalization-of-a-mind-boggling-box-opening-puzzle#comment1252164_480917
  [3]: https://oeis.org/A062868