Hint/comments for 1:

There a few steps to take here. Start by considering a side view, so the truncated cone (also called a frustrum) is a trapezoid and the sphere is a circle. Note extending the sides of the trapezoid forms a triangle. Try to use similar triangles and tangent lines of the circle to proceed from here.

Hint/comments for 2:

Since the centers of the cube and sphere are the same, the sphere will intersect each edge of the cube in the same way. If 1/2 of the each edge is inside the sphere, doesn't this mean there are pieces of 1/4 of the edge on each end? Try to draw a picture and consider some right triangles from here.