What I am wondering is if the Hirsch conjecture has a simple proof (just a few lines) in $3$ dimensions, perhaps by using Steinitz's theorem or Kuratowski's theorem and some kind of induction argument.
For $d=3$, the Hirsch conjecture states that for a $3$-polytope $P$, the diameter $\delta$ of the graph of $P$ is bounded in terms of the number of faces $F$ by the inequality $\delta \le F - 3.$
Note: in higher enough dimensions $d$ the Hirsch conjecture fails, as was recently shown by Francisco Santos (see http://annals.math.princeton.edu/articles/3941), but it is true for small $d$.