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]
<br />
<hr>
After seeing your addendum on motivation, perhaps you should investigate the
literature on packing disks on a sphere.
For example, you might explore
Neil Sloane's web pages on
"[Nice arrangements of points on a sphere in various dimensions][4]"
which address the problem of 
> placing $n$ points on a sphere in $d$ dimensions so as to maximize the minimal distance (or equivalently the minimal angle) between them. 

For example, here is the best packing known for $n=50$, due to Székely (1974):
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
![50 points on sphere][5]
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<sub>[Image © Hugo Pfoertner, 2001][6]</sub>


  [1]: http://en.wikipedia.org/wiki/Vertex-transitive_graph#Finite_examples
  [2]: http://mathworld.wolfram.com/GeodesicDome.html
  [3]: https://i.sstatic.net/fEsSg.jpg
  [4]: http://www2.research.att.com/~njas/packings/
  [5]: https://i.sstatic.net/uZ5em.jpg
  [6]: http://www.enginemonitoring.org/sphere/index_4.htm