Ways to look at a polyhedral graph Motivation
There are at least three interpretations of an abstract polyhedral (= planar 3-vertex-connected) graph:

*

*the 1-skeleton of a convex polyhedron (when embedded into $\mathbb{R}^3$)


*a polygonization of the sphere (when embedded into the sphere $\mathbb{S}^2$)


*a polygonization of a polygon – for each of its faces (when embedded into the plane $\mathbb{R}^2$)
In any case there are many geometric realizations:

*

*of a polyhedron


*of a polygonization of the sphere


*of a polygon and its polygonizations
Question
I'd like to understand in an abstract setting:

What do these interpretations and realizations have to do with each
other?

Example
For many (but not all) polyhedral graphs there is a realization as a convex polyhedron that is inscribable into the sphere. A central projection of this polyhedron back onto the sphere induces a polygonization of the sphere.
Taken for granted is Steinitz' theorem. The question is not about this.

EDIT: For completeness' sake I should mention embeddings of a polyhedral graph into:

*

*the hyperbolic space $\mathbb{H}^2$


*the 3-dimensional sphere $\mathbb{S}^3$


*the hyperbolic space $\mathbb{H}^3$
 A: This is not a complete answer, just a pointer on one aspect of your question,
the relationship between two of your three "interpretations."
As you note, not all polyhedral graphs have
a realization that is inscribable in a sphere.  Here is an explicit example:

"Uninscribable 4-Regular Polyhedron."
  Electronic Geometry Model No. 2003.08.001.
  David Eppstein and Michael Dillencourt.
  (Link to abstract, and to the images below)


           

    


There is a characterization of the inscribable polyhedral graphs, obtained in 1992:

Craig Hodgson, Igor Rivin, and Warren Smith.
  "A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere."
  (arXiv link)


Now that I understand better the thrust of the question from the OP's comment below,
I believe this recent survey by Günter Rote should be helpful:

"Realizing Planar Graphs as Convex Polytopes."
  Graph Drawing.
  Lecture Notes in Computer Science Volume 7034, 2012, pp 238-241.
  (Springer link)

