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?

You're right, but still a couple of errors were left (see my later correction).  As I originally had mentioned there is much opportunity for error in this puzzle.

The way I found out my error was I tried sequences predicted by my tetrahedron and cube formulae in the online encyclopedia of integer sequences.  The tetrahedron formula proved correct as the sequence of numbers led to that description, but the cube sequence was not found.  The tetrahedron sequence description led me to Martin Gardner's New Mathematical Diversions from Scientific American, page 246, where a number of formulae are shown, including also the one for cubes, but given as (25n^4-120n^3+209n^2-108n)/6.

Plugging numbers in and looking up the sequences on the online encyclopedia led to http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A006529, which indicates that the formula given in the first edition of Gardner's book was wrong, and led me to the corrected version at http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A047780, where I saw that two of my coefficients were wrong.

It was the devil of a time trying to find where those extra amounts came from.  So we're in good company, if Martin Gardner originally got it wrong.