If a right circular cone intersects a plane that runs parallel to the cone's axis but does not pass through its vertex, the resulting curve is a pair of hyperboles.
If a right circular cone intersects a plane that runs perpendicular to the cone's axis but does not pass through its vertex the resulting curve will be a circle.
No but it has a circular edge
cone
It is a cone.
An Ellipse
It will be a hyperbola.
hyperbola
hyperbola
A line is produced
if a right circular cone intersects a plane that goes through both nappes of the cone, but not through the vertex, the resulting curve will be a hyperbola
Two nappes of a right circular cone meet at a point called the vertex.
A point.
If I understand your description correctly, a line.
It is the vertex. It can also be called the apex but that has a connotation of being the top-most point and so is not appropriate when both nappes are concerned.
The intersection will consist of only one point.
Acute angles.