For problem 27 here is a view of the cube and the tetrahedrons that are removed, and also a view of the created solid $\mathcal{S}$

and here's a view that lets us see the diagonal joining the two vertices from which the tetrahedra originate.

For problem 30, to see that the intersection of a sphere and frustrum create a circle: imagine you have a sphere and a circle with radius smaller than that of the sphere; then the sphere would not be able to go through the circle and would get stuck touching whole circle (you can literally try this with say a soccer ball for the sphere and a coffee mug for the circle). In the case of a cone the same thing would happen, as all the points of the sphere that are in the cone lie on the same plane parallel to the base of the cone.