What happens to the shape of a pyramid as the number of faces increases?

Answer 1

It depends on whether or not the pyramid base remains convex or if whether some of the additional sides created ridges.

If the base remains convex, sections of the pyramid base will become more like a circle and the sloped surface above that part will become smoother.

It would be wrong to conclude that the pyramid would become more like a cone because it is entirely possible that all the additional sides - possibly infinitely many - are all between two adjacent vertices of the original base polygon. You would then have a base that is a segment of a circle - a third, or a quadrant, etc..