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Problem : How does the length of the altitude of a cone compare to the length of any of the segments that compose the lateral surface of the cone?

The altitude, because it is perpendicular to the plane of the base, is always of lesser or equal length than the segments that make up the lateral surface.

Problem : When is a cone a pyramid?

When the base is a polygon

Problem : Does the altitude of a pyramid have to be in the interior of the pyramid?

No. The altitude can lie either in the pyramid, outside of the pyramid, or even on the pyramid. The altitude lies on the pyramid when the vertex can be connected to the polygon with a line perpendicular to the plane of the base. It all depends on where the vertex is fixed.