Is Z a random variable of X? Is Z a random variable of X if it is transformed from X? If Z is a random variable, and therefore a function, of X, could one denoted Z as Zₓ(x)?

Let's flip it around for clarify.
If $X$ is a random variable, then all the common transformations of $X$ are also random variables.  However, these transformations likely do not have the same distribution as $X$.  For example, if $Z = e^X$ and $X$ is normal distributed, then $Z$ is log-normally distributed (literally, $\ln(Z)$ is normal).  Other transformations have other relationships.
Note, some functions of $X$ do not give a random variable.  For example, $\frac{X}{X}$ would of course always equal 1 (when $X\neq 0$).
As for notation, you can just use $Z(X)$.  No need to the subscript.  Although, if if you have several variables to transform, e.g. $X$ and $Y$, then the subscript could be useful when you are evaluating specific values: e.g. $Z_X(4)$ vs $Z_Y(4)$.
