It is possible to label the vertices of a cube with strings of 3 binary digits in such a way that two vertices are adjacent if and only if the correspondent strings differ one from the other in exactly one digit position.
Is there any similar labelling for an arbitrary polyhedron? (I'm especially interested in closed polyhedra.) The main feature of the labelling I am requiring is that it easily indicates adjacency relation (whether or not two vertices are adjacent), and, if possible, "facy relation" (whether or not three or more vertices are all the vertices of a single face).
You should look into zonotopes, which are Minkowski sums of line segments (or projections of cubes, it turns out the two are equivalent). I'm fairly sure that all zonotopes can be represented this way, although I'm not positive. For example, a hexagon is a zonotope. If we project your cube onto a surface perpendicular to the line through $000$ and $111$, then the six exterior vertices will be those of a hexagon, retaining the original labels, which would still tell you about their adjacencies.
I wouldn't be surprised if such a labeling turned out to be unique to zonotopes, although I'm far less sure of this (but I suspect it's true).
Ziegler's book Lectures on Polytopes is a good reference for zonotopes.