How much of the planet's surface can Az'glqssk see if she is in the middle of one of the faces? What about on one of the edges? Or at a vertex?

It may help to use this Schlegel diagram: it is a stretched net of the dodecahedron: you can see 11 faces, and you can think of the space round the outside as being the twelfth face, as it joins the five round the outside. So you can "see" all the edges and vertices, and what's connected to what.

Of course, if you want to calculate any distances, you need to remember that those pentagons are actually regular!