Is the median preserved for any strictly monotonic mapping? Problem
Let $X\sim f_X$, where $f_X$ is the probability density function of $X$. Let $g: \mathbb{R} \to \mathbb{R}$ a strictly monotonic (decreasing or increasing) mapping. I aim to prove or disprove:
$$
g[\mathrm{median} (X)] = \mathrm{median}[g(X)]
$$

Try
Let $x_0 = \mathrm{median} (X)$, $Y=g(X)$, $y_0 = g(x_0)$. Then, we want to show:
$$
0.5 = \int_{-\infty}^{y_0} f_Y (y) dy
$$
WLOG, assume $g$: increasing. By change of variables, $f_Y(y) = f_X(x)/g'(x)$ and $dy=dx g'(x)$. Thus,
$$
\begin{align}
\int_{-\infty}^{y_0} f_Y (y) dy &= \int_{-\infty}^{y_0} f_X(x) dx \\
&\neq \int_{-\infty}^{x_0} f_X(x) dx = 0.5
\end{align}
$$
So, if $f_X>0$ a.s., then the above statement holds only if $g(x_0) = x_0$. However, take an example of log-normal distribution, where $X\sim N(\mu, \sigma^2)$ and $g(x)=\exp(x)$. There is no real solution to $g(x_0) = x_0$, and $f_X >0$ a.s. But it is still true that $\mathrm{median}[g(X)] = g(\mu)$.
So my disproof is invalid? Any suggestions are welcome.

The conjecture is true and your disproof is flawed.  The flaw in your steps occurs when you make a change-of-variables in the initial step but do not change the range of integration accordingly.  While it is not the simplest method of proof (see here for a simpler approach using the CDF), you could construct a proof of this result using the density functions along the lines you are trying.  If $g$ is strictly increasing and differentiable you have $g'(x) \geqslant 0$ for all $x \in\mathbb{R}$, which then gives:
$$\begin{align}
\int \limits_{-\infty}^{y_0} f_Y(y) \ dy 
&= \int \limits_{-\infty}^{g(x_0)} f_Y(y) \ dy \\[6pt]
&= \int \limits_{-\infty}^{x_0} f_Y(g(x)) \cdot \frac{dy}{dx} \ dx \\[6pt]
&= \int \limits_{-\infty}^{x_0} f_Y(g(x)) \cdot g'(x) \ dx \\[6pt]
&= \int \limits_{-\infty}^{x_0} f_X(x) \ dx \\[12pt]
&= \frac{1}{2}. \\[6pt]
\end{align}$$
When proving this result, I would also recommend you tighten up your assertion that you can use increasing $g$ without a loss of generality.  You need to make it clear why the decreasing case is analogous to the increasing case.  This is related to the formula for transformation of random variables for monotonic transformations.
