My bad, I got way too confident

I mean, this is a simplicial graph, and $\pi_1$ of graphs are free.

$X$ is not path-connected, and any path-connected component is a increasing union of grids in finite dimension, $\pi_1(X)$ is just going to be a free group of countably infinite rank (as for the grid in $\mathbb R^2$).

@Ville I guess the question is about Cayley graphs w.r.t. infinite generating sets.

Do you mean “direct sum of all proper subgroups” ?

See doi.org/10.1017/S144678870000121X, in particular in the second section for an infinite family.

You should put a restriction on the rank, as shown by the example of Dave Rusin. Even with that, I don’t believe there is a classification, the size of the largest finite $2$-generated group of exponent $n$ probably grows much faster than exponential in $n$ (see « restricted Burnside problem »).

If two adjacent cars "decide" to move on a given step (with a free spot on the right), do they both move or just the right one?

Just isomorphic to its dual as an abstract polytope, hypercube and hyperoctahedron coincide in dimension $d=1,2$.

In this case, it is algebraic for $d=1$, and D-finite (but not algebraic) for all $d\ge 2$.

I guess you should have a look at « Knapsack problem in groups » (where we usually take $x_i$ non-negative integers). It has been studied in nilpotent groups, see arxiv.org/abs/1507.05145 and arxiv.org/abs/1606.08584 , but I don’t think the specific result you’re asking for was written down beforehand.

From an algorithmic point of view, there is this paper arxiv.org/pdf/math/9811012, but it’s not quite as explicit as what you were asking for.