I know the hints will be less useful now, but thanks for your patience.

For 9.22, try to consider a similar argument to Example 9.2. That is, look at the diagonal cross section.

For 9.25, remember we can create a regular tetrahedron using face diagonals of a cube. The key here is that, by symmetry, if a regular tetrahedron fits inside the sphere, so does this corresponding cube. Thus, it is enough to start by looking for the largest cube that fits inside the sphere.

For 9.28, I discussed this in an earlier post here with a diagram (this is a little more than just a hint, but hopefully it helps visualize): https://classes.areteem.org/mod/forum/discuss.php?d=1077