Is this SLOPE of the line even possible? Is it even possible to build a line having slope of $3$? Could it be a mistake?

 A: Not a mistake.  You seem to have a very basic misunderstanding.  The line you drew passes through the TWO points (-2,0) and (0,-8). Its slope is -4. That's
just one problem.
A: Slope of a line is nothing but the tangent of angle which the line make with $x$-axis or simply speaking $\tan\theta$.
Since value of $\tan\theta$ vary from $\infty$ to $-\infty$ so yes a slope of 3 for a line is possible.
Now, you may have the question that How to construct a line having slope 3? Since you have the point (-2,-8), just plot it on cartesian axes and extend a line from that point such that on moving along line you will move a distance of 3 units on $y$-axis for a distance of 1 unit on $x$-axis.
A: A slope of $3 = \frac{3}{1}$ means that starting at any point on the line (like say $(-2,-8)$), if you move $3$ units up and $1$ unit right then you'll get another point on the line.

So, yes.  It is possible.
My advice is to try playing around with different points and slopes on WolframAlpha to get a feel for it.
A: You should think of a line with its general form of equation.
$ax + by + c = 0$
Then you can represent a vertical line when $b = 0$
The same goes for horizontal ones when $a = 0$
The only constrains come if $a = 0$ and $b = 0$, then it does not represent a line any more.
