The answer to the third question is **no.**  This is a rather counter-intuitive discovery of Micha Perles from the sixties.  See this [paper](http://arxiv.org/abs/0710.4453) of Ziegler, for a simpler construction and other pertinent information.  However, for polytopes in dimension $3$, the answer is **yes**, as mentioned in the same paper of Ziegler.