In the solution, it says for $x \leq 0$, it's $P(X \leq x)$, but shouldn't it be $P(X \leq 0)$ or $P(x \leq X \leq 0)$? Also, on that same line, how did it go from $e^{-|y|}$ to $e^{y}$?
And for the second line, why is $F_X(0)$ there, but it wasn't there for $x \leq 0$?
In the solution, it says for $x \leq 0$, it's $P(X \leq x)$, but shouldn't it be $P(X \leq 0)$ or $P(x \leq X \leq 0)$?
As sudeep mentioned you are missing the definition of CDF.
The cumulative distribution function is the probability that the variable takes a value less than or equal to x. That is
$F_X(x)=P(X \leq x)$
Also, on that same line, how did it go from $e^{-|y|}$ to $e^{y}$?
Definition of absolute function says
$|y|= \begin{cases} -y & -\infty < y < 0 \\ \ \ y& \ \ 0 \leq y < \infty \\ \end{cases}$
substituting $-y$ gives $e^{y}$
And for the second line, why is $F_X(0)$ there, but it wasn't there for $x \leq 0$?
Again using the definition of CDF which gives probability less than or equals $x$. When you are computing CDF for $x>0$ you have to add the probability for $x<0$. When we were calculating $P(X<x)$ for $-\infty<x<0$ there was nothing to add because $F(-\infty)=0$.
Does it help?