Suppose we take a convex polyope $P$ and a face $A$ with vertices $a_1,\ldots, a_n$. We hold the polytope with $A$ flush with the surface and slowly lower it, keeping $A$ parallel to the surface thoughout. We continue lowering until the water level reaches some vertex $b_1$ not belonging to $A$. Then let $b_1,\ldots, b_m$ be all the vertices at the water level. I wonder:
Is every $b_i$ joined by an edge to some $a_i$?
Seems physically obvious. But so do many facts about polytopes, such as the linear-inequalities/convex-hull definitions being equivalent.
If you consider the part of the polytope between the water level and the plane spanned by $A$ you get a smaller polytope $Q$. This $Q$ has all $a_i,b_j$ as vertices but might have extra vertices created when edges of $A$ pass through the water. Nevertheless all the vertices are contained in one of the two planes. This suggests the following perhaps easier question.
Suppose $P_1,P_2$ are two parallel planes, and $P$ is a polytope whose every vertex is in either $P_1$ or $P_2$. Is each vertex in $P_1$ joined by an edge to a vertex of $P_2$?
The answer to your second question is Yes (and so is the answer to the first).
In general, for every vertex of a (full-dimensional) polytope $P\subset\Bbb R^d$, the directions of the edges incident to that vertex span the whole $\Bbb R^d$.
If a vertex in $P_1$ would have edges only to other vertices in $P_1$, then the span would be of dimension $\le \dim(P_1)= d-1$, hence, not all of $\Bbb R^d$.