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Probably not: there is a 4-dimensional (infinite) contractible simplicial complexes with a simplicial action of a cyclic group of order $pq$ that has no global fixed point, for any distinct primes $p$, $q$.
Being able to watch, while entering the conference centre, Keizo Ushio either sculpting with his power tools or standing thinking was one of the highlights of the 2006 ICM.
There is one cube of each dimension $\geq 0$, but all of the boundary maps are trivial, so the unreduced cubical homology of a point is one copy of $\mathbb{Z}$ in each dimension.
In general, higher differentials are higher cohomology operations, defined only on the kernels of the differentials below them. If $a$ is a mod-2 cohomology class with $Sq^1(a)=0$, this says that the coboundary $b$ of an integral lift of $a$ is divisible by~2. The secondary operation in this case looks at $b/2$ as a mod-2 cocycle; this is defined only because $b$ was 0 mod-2.
If you were prepared to change generating set to the $n-1$ elements $(1,2),(2,3),(3,4),\ldots,(n-1,n)$ each of order two there is a very good geometric theory of word length with respect to these generators. The thing to search for is finite Coxeter groups: the symmetric group $S_n$ with these generators is the Coxeter group of type $A_{n-1}$. For example, there is a unique element of $S_n$ that has maximal length with respect to these generators (although there are many words of that length that represent it). The geometry behind this is that the antipodal point on a sphere is unique.
