I assume you are aware that the vertex-transitive graphs represent a relatively constrained list? See this description. 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?
They fail to satisfy your criteria, but they get perhaps
as close as is feasible, and may serve, depending upon your application:
Geodesic Dome http://cs.smith.edu/%7Eorourke/MathOverflow/GeodesicDome.jpg
