1. How many different tetrahedrons can be produced by coloring each face a solid color and using n different colors? (Two tetrahedrons are the same if they can be turned and placed side by side so that corresponding sides match in color.)

2. How many cubes with n colors?

"This adds two cases, each of which is multiplied by C(4,2) (even within the C(n,4) coefficient), so we're adding 12 to the coefficient of C(n,4).  I had previously reported that coefficient as 60, but as you can check, it really should have been 56--my arithmetic was wrong.  So the new total should be 68."

Actually, it only adds one case.  As I noted in my earlier response (actually, I also only noticed this after your second post, and marked the part that I edited accordingly), the other case you missed is in the situation where one doublet is adjacent and one is opposite.  You correctly note that this leads to one possibility, but forgot that there are two ways to pick which doublet is adjacent and which is opposite.