Say I have $f:\mathbb C\rightarrow \mathbb C$, so $$ f(x+iy)=f(z)=u(x,y)+iv(x,y) $$
Q1: Does it mean the functions $u$ and $v$ are \begin{align} u,v:\mathbb R^2\rightarrow \mathbb R \tag{1} \end{align} Or \begin{align} u,v:\mathbb C\rightarrow \mathbb R \tag{2} \end{align}
Q2: The notation $u(x,y)$ and $v(x,y)$ implies that $$u,v:\mathbb R^2\rightarrow \mathbb R$$
But if $u,v:\mathbb C\rightarrow \mathbb R$, shouldn't we instead write $u(z)$ and $v(z)$?
The functions $u,v$ obviously have two real arguments and take a real value. Hence $\mathbb R^2\to\mathbb R$. This is why you can't write $u(z),v(z)$.
In some sense, there is an implicit function $\iota$ that decomposes a complex number in a pair of real, imaginary components,
$$\iota:\mathbb C\to\mathbb R^2:\iota(z)\to(x,y)=(\Re(z),\Im(z))$$ and
$$f(z)=u(\iota(z))+iv(\iota(z)).$$