The phrasing of the exercise is a bit strange for parts (a) and (d). If you take the derivative of
##*F = q \sin\theta d\theta \wedge d\phi##, you get ##d*F = q \cos\theta d\theta \wedge d\theta \wedge d\phi = 0##. So (a) and (d) become trivial.
For (b), you made a mistake. ##F_{tr}## should be...