Why does a ray parallel to principal axis passes from focus after reflection from a concave mirror and vice a versa is also true? Why is it so? I read this as basic rules or laws you can say in REFLECTION from concave mirror.
 A: Consider an incoming ray which is parallel to the principal axis which hits a concave mirror at $X$.

The normal to this mirror at $X$ passes through the centre of curvature of the mirror $C$.
$\frac {h}{CP'}= \tan \alpha, \, \frac {h}{FP'}= \tan 2\alpha, \, $
For small $\alpha$ ie incoming ray close to the principal axis
$CP' \approx CP,\, FP' \approx FP,\, \tan \alpha \approx \alpha,\, \tan 2\alpha \approx 2\alpha$ where $P$ is the pole of the mirror.
$\Rightarrow CP \approx 2 FP$
$CP$ is a property of the mirror and $FP$, the focal length of the mirror, is thus (approximately) independent of the angle $\alpha$ as long as $\alpha$ is small.
If $\alpha$ is not small then a mirror defect called spherical aberration occurs as shown below.

(On Desmos)
You often see this whilst having a drink with the bright line being called a caustic curve.

A: A spherical concave mirror does not have a single focus. The focal length depends on the point of reflection. The point at which a light ray, initially parallel to the principal axis intersects the principal axis is $\textbf{defined}$ to be the focus. As the point of reflection approached the center of the lens, the focal length converges to half of radius of the mirror. This can be proved with geometry.
However for a concave parabolic mirror, all light rays parallel to the axis pass through a single focus after reflection and this can indeed be proved  (you can find the equation of normal to the parabola at a point and then use the laws of reflection to prove this)
A: As you know the image of an object, whose distance from mirror is infinite, form at focus and you must have observed that rays from infinity are all parallel to principal axis. Now your question, if rays are parallel to principal axis then they must pass through focus because they are same as rays coming from infinity. 
Hope it helps.
A: Consider any lens or mirror which is part of a sphere. The focal length is half the radius. 
Next consider any point on the lens or mirror. The line joining this to the center of curvature is obviously perpendicular to the tangent at that point. 
Any line joining the focus to this point and one parallel to the principle axis make equal angles with the line joining the center of curvature. 
Since angles of incidence and reflection are equal, the rays parallel to the principal axis pass through focus and vice versa. 
A: If you could relate this situation to a parabola, where every point on the parabola is at the same distance from the focus as it is to the directrix. So it seems pretty natural that the focus is the point where all those light rays meet when they reflect off curved curved mirrors.
