I assume you are aware that the vertex-transitive graphs represent a relatively constrained list?
See [this description][1].  I fear that adding your conditions (2,3,4) on top of
vertex-transitivity will
again confine you to the skeletons of the Platonic solids.

Are you familiar with Fuller's [geodesic domes][2]?
They fail to satisfy your criteria, but they get perhaps
as close as is feasible, and may serve, depending upon your application:
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
![Geodesic Dome][3]


  [1]: http://en.wikipedia.org/wiki/Vertex-transitive_graph#Finite_examples
  [2]: http://mathworld.wolfram.com/GeodesicDome.html
  [3]: http://cs.smith.edu/~orourke/MathOverflow/GeodesicDome.jpg