Why do most of the operations treating $dy/dx$ as a ratio works?

When Leibniz designed the notation he was thinking of $\frac {dy}{dx}$ as a ratio of infinitesimals. But, the analysts of the 18th century were unable to develop this into a rigorous theory, and infinitesimals were replaced by limits.

And $\frac {dy}{dx}$ no longer a ratio.

However, it does work like a ratio for just about all of elementary calculus.

When you get multivariate calculus and partial differentiation, the thought of the differential operator as a ratio really starts to break down.

the total derivative: $\frac {d}{dt} f(u,v) = \frac {\partial f}{\partial u} \frac {du}{dt} + \frac {\partial f}{\partial v} \frac {dv}{dt}$

And as the math becomes increasingly advanced the differential operator behaves less and less like a ratio.

Note, that in the counterexample you gave, $F$ is a function of $y$ and $x$. Thus, $\frac{\partial F}{\partial y}$ is a shorthand for $\lim\limits_{h \to 0} \frac{F(x,y+h) - F(x,y)}{h}$ which is very much different than $\lim\limits_{h \to 0} \frac{F(x+h,y) - F(x,y)}{h}$.

For this reason you can't cancel $\partial F$ against each other, thus:

$$\frac{\partial F/ \partial y }{\partial F/ \partial x} \neq \frac{\partial x}{\partial y}$$

Here's a nice example: Consider the equation $xyz=1.$ Then

$$\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x} = -1,$$

even though cancellation of the differentials suggests otherwise.