Let say in x,y dimension if a line cross over the x-axis instead of y (as shown in image above, unlike y=mx+c ) then how equation will change or will the equation has any impact apart from the x-intercept in this case? Is this a valid case?

One way to write an equation of an arbitrary line in the $x,y$ plane is

$Ax+By+C=0.$
If $A=0$ you get a line parallel to the $x$ axis; if $B=0$ you get a line parallel to the $y$ axis.
If you set $A=m,$, $B=-1,$, and $C=b$ then the equation $Ax+By+C=0$ describes the same line as $y=mx+b.$.
But people often are interested in the equation $y=mx+b$ for reasons other than the shape it describes in a plane. We may have some quantity we can either control or observe taking different values, which we'll represent by the name $x$, and some other quantity, which we'll call $y$, whose value has some relationship to the value of $x.$
The relationship $y=mx+b$ is one of the simplest possible kinds of relationship that can occur under these circumstances. And it happens also to be possible to visualize a relationship like this by plotting a line on a graph.
A vertical line can not be the plot of such a relationship, because the first thing we wanted to see was a variety of different values of $x$, and the vertical line has only one $x$ value. The fact that $y=mx+b$ cannot describe a vertical line therefore is irrelevant to the study of these kinds of relationship.

I think if the line is parallel to y - axis (as shown by the image you have attached). Then the slope of that line will not be defined because tan90 is not define .
If only slope is not define then the question of repsenting that in slope -intercept form doesn't make any sense.