Are edge-vertex graphs of polytopes (decent) expanders?
For 0/1-polytopes (all vertex coordinates are 0 or 1), this is not known to be true.  There is a conjecture by Mihail and Vazirani that the edge expansion of the graph of a 0/1-polytope is at least one.  More information is described in a paper by Volker Kaibel.

I should note two things.  (1) For 0/1-polytopes, the Hirsch conjecture is true.  (2) When performing a random walk on the vertices of a polytope, we need to take care of possible degeneracy.  One vertex can correspond to a lot of bases, and so the walk can stay at the same vertex if we perform a random walk over the bases.  If we want to perform a random walk over the vertices, we need to have a procedure that gives a random adjacent vertex.