Otherwise neat and hope you find a good internship.
Probably the easiest fix for the controls would be to limit which mouse movements move the shape: moving down would spin it up, but moving up would do nothing. Moving left would spin it right, but moving back across it would do nothing. That way you could move it into place and then come back to click on where you want.
Also, it would be lovely if the most centered face, the one that is changeable, became outlined when it comes into focus.
EDIT: Oh and add a histogram of how far along a solution you are
I could only google a book called "Map Coloring Polyhedra and the Four Color Problem". Unfortunately, it's not available to read online. From what I've found, it seems to me that there is only a handful of proven facts about colorings of some polyhedra.
[ADD]: four color theorem works for every planar graph and it looks like one can make such graph corresponding to any convex polyhedron, so it seems that original OP statement holds.
I first thought, "That's exactly what I said," and then I realized that "homeomorphic" isn't exactly a household word.
As for the control, here is what I would do:
Imagine a circle on the center of the screen that stopps the spin, outside the circle, rotate with a speed proportional to the distance from the center.