It is called a frustrum
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Yes if cut in half from its apex to its base diameter.
A cylinder and a cone have infinitely many planes of symmetry because of the circular face. However, a cylinder can also be cut in half lengthwise (imaging cutting a soda can in half), while a cone cannot (imagine cutting a ice cream cone in half). Therefore, a cylinder has one more plane of symmetry than a cone.
hemisphere
Depends on the way you cut the cone, but the outline is either an ellipse or a parabola.
A frustum is a truncated cone or pyramid. In other words, a cone with the top cut off, much like an ice cream cone with a flat bottom.