If $S_1=0$ and $S_2=0$ represent two non intersecting circles, what does $S_1+\lambda S_2=0$ represent?
I know that $S_1+\lambda S_2=0$ represent the family of circles that are coaxial with $S_1=0$ and $S_2=0$, (where $\lambda\ne -1$, because then it would become the radical axis) but I cannot see it geometrically. If the circles are intersecting, then it is clear that $S_1+\lambda S_2=0$ represents the family of circles passing through their common points of intersection and hence they would have the same radical axis, that is, their common chord. How do I prove that for non intersecting circles?
Secondly, I want to ask if we are given a line $L=0$ and a circle $S=0$, what does $S+\lambda L=0$ represent if they are non intersecting? For intersecting case, it is clearly that it would be family of circles passing through their points of intersection. I think it would represent the circles $S'=0$ such that $S=0$ and $S'=0$ have the radical axis $L=0$, but there is no intuition, because intersection points are imaginary.
Maybe it is just that everything is right in front of me, but I am not able to understand it clearly. I would prefer a complete answer rather than comments.
Your two questions are in fact one, because the pencil for circles $C_1+\lambda C_2$ contains the radical axis of $C_1,C_2$ (the one that you rejected), and conversely, the pencil $C+\lambda L$ defined by a circle and a line is such that $L$ is the radical axis of $C and another circle in the pencil.
WLOG, consider
$$\begin{cases}S_1\equiv x^2+y^2=1,\\S_2\equiv(x-d)^2+y^2=r^2.\end{cases}$$ By subtraction, the line in the pencil is
$$L\equiv d(2x-d)=1-r^2$$ or
$$x=\frac d2+\frac{1-r^2}{2d}.$$
The distances of the intersection with the axis $x$ to the centers are
$$\pm\frac d2+\frac{1-r^2}{2d}$$ and we do have by Pythagoras
$$\left(\frac d2+\frac{1-r^2}{2d}\right)^2-1^2=\left(-\frac{d}2+\frac{1-r^2}{2d}\right)^2-r^2,$$
which proves that the lengths of the tangent segments to the circles from this point have equal lengths and the line is the radical axis.
https://en.wikipedia.org/wiki/Radical_axis#/media/File:Radical_axis_orthogonal_circles.svg
These families of circles are well illustrated by the Appolonius' coordinates system: