Analyticity of $\overline {f(\bar z)}$ given $f(z)$ is analytic

Alternatively, you can use that $z\mapsto \overline{z}$ is continuous to make: $$\begin{align} \lim_{h \to 0} \frac{\overline{f(\overline{z+h})}-\overline{f(\overline{z})}}{h} &= \lim_{h \to 0} \frac{\overline{f(\overline{z}+\overline{h})}-\overline{f(\overline{z})}}{\overline{\overline{h}}} \\ &= \overline{\lim_{h \to 0}\frac{f(\overline{z}+\overline{h})-f(\overline{z})}{\overline{h}}} \\ &= \overline{\lim_{\overline{h} \to 0}\frac{f(\overline{z}+\overline{h})-f(\overline{z})}{\overline{h}}} \\ &= \overline{\frac{\rm d}{{\rm d}z}\bigg|_{z=\overline{z}}f(z)}\end{align}$$

A function is analytic if its real and imaginary parts (your $u$ and $v$) are differentiable, and those derivatives satisfy the Cauchy-Riemann equations. The $u$ and $v$ are real-valued functions of two real variables, so “differentiable” in this case just means differentiable as functions from $\Bbb{R}^2$ to $\Bbb{R}$; there's no further complex analysis to do.

So you just need to worry about whether the functions $(x,y) \mapsto u(x,-y)$ and $(x,y) \mapsto -v(x,-y)$ are differentiable for any real-differentiable functions $u$ and $v$. Which should be pretty clear, I think...

When you differentiate a $C^1$ function $f(x, y)$ to check whether the CR equations hold, you are computing real (partial) derivatives; (complex) analyticity is the same as existence of the complex derivative. So yes, your method is correct.