Voronoi Diagram: Exactly 2n-5 vertices
You made a small mistake by thinking that the Voronoi diagram is a planar graph. Remember that it has those rays that go to infinity. you can make a planar graph out of the Voronoi diagram by adding a new vertex and making all rays incident to this "apex". Plugging this into Euler's formula yields

$$ 2n-4 - e + n = 2 \quad \Rightarrow \quad  e= 3n - 6.$$

A planar graph has $3n-6$ edges if it is a triangulation (this can als be obtained via Euler's formula – just set $3f=2e$). As a consequence the dual graph to your Voronoi diagram (the Delaunay tessellation) is a triangulation. Or phrased differently, every vertex of your Voronoi diagram  has degree~3 and there are three rays.