conic sections
According to Wikipedia,
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
However most of the images actually show a double cone instead of a cone and it makes sense to me since a hyperbola has two components.
So is it true that despite its name and definition, conic section is actually an intersection of a plane with a double cone?
Converting comments to answer, as requested:
The Dandelin spheres answer question (1): a focus of a conic section is the point of tangency of its plane with one of those spheres. Clearly, the point on tangency lies on the cone axis if and only if the plane is perpendicular to that axis; therefore, the axis contains a focus in, and only in, the case of a circle. The axis point has some relation to the conic section, but not one as interesting or useful as a focus. (As for "Does the diameter of the base of the cone pass through the focus of the hyperbola?": note that the cone has no base. The thing extends, and expands, infinitely-far.)
This notion of points seeming to move "out to infinity and come back on the other side" is not uncommon in analytical geometry. (It's kinda what happens along asymptotes, when you think about it.) You might be interested in studying Projective Geometry, which adds a "line at infinity" to the standard Euclidean plane; this broader context helps unify all the conic sections into a single kind of curve that has various different relations to that line. (The parabola, in particular, has its "second vertex" on that line.)
Viewing a vertical hyperbola in a direction perpendicular to the cutting plane, the "edges" of the cone appear as the hyperbola's asymptotes.
Consequently, the diameter of the circle in question is simply how "wide" those asymptotes are at the level of the hyperbola's vertex. That is, if $A$ is a vertex of the hyperbola with center $O$, and $A^\prime$ is the point where the perpendicular to $\overline{OA}$ at $A$ meets an asymptote, then $|AA^\prime|$ is the radius of the circle. This distance is precisely the length of the conjugate semi-axis.
Writing $a$, $b$, $c$ for the transverse semi-axis, the conjugate semi-axis, and the distance from center to focus, and writing $e$ for the hyperbola's eccentricity, we know that $$a^2 + b^2 = c^2 \quad\to\quad b^2 = c^2 - a^2 = a^2\left(\frac{c^2}{a^2}-1\right) = a^2 \left(e^2-1\right)$$
So, the horizontal circle that meets a vertical hyperbola at its vertex has radius $b = a \sqrt{e^2-1}$. $\square$
Best Answer
There's a very general definition of "cone" which is the following.
Let $\cal C$ be a (closed, regular) curve in $\Bbb R^3$ and let $P$ a point not in $\cal C$. Then the cone through $\cal C$ with vertex $P$ is the ruled surface made of all lines $QP$ as $Q$ varies in $\cal C$.
When $\cal C$ is a circle with center $C$ and $P$ is chosen so that $PC$ is orthogonal to the plane containing $\cal C$ you get what you call "double cone".
You really need double cones if you want your section to be an hyperbola, which is made of two disconnected parts, lying on different "parts" of the double cone.
General cones are a basic example of flat surfaces (they can be flattened out isometrically on a plane).
Also note that if you let $P$ move far away from $\cal C$ the lines $QP$ tend to become parallel and will if you let $P$ "go to infinity". In this sense cylinders are degenerate cones (the vertex of a cylinder being "at infinity")