Which kind of functions $f(x, \ y)$ can be written as $g(x)\cdot h(y)$? What are the properties of the function $f(x, y)$ that make it possible to be separated into a product of two one-variable functions, $g(x)$ and $h(y)$?

Let's borrow logarithm for a minute (i.e. assume that it is well-defined everywhere).
Then, if $f(x,y) = g(x) h(y)$, then $\ln f(x,y) = \ln g(x) + \ln h(y)$, so $(\ln f)_{xy} = 0$, i.e. $\left(\dfrac{f_x}f\right)_y=0$, i.e. $\dfrac{f_{xy}f-f_xf_y}{f^2} = 0$, i.e. $f_{xy}f=f_xf_y$.

For the other direction, assume $f_{xy} f = f_x f_y$. Pick $(x_0, y_0)$ such that $f(x_0, y_0) \ne 0$ (otherwise just set $g=h=0$). Then define $g(x) = f(x,y_0)$ and $h(y) = \dfrac{f(x_0,y)}{f(x_0,y_0)}$.
I don't know if this works.
