Are derivatives defined on functions or variables? My question is, is differentiation done on functions or on variables/values?
For example take $z=f(x,y)$, in this case we can 'fix' one of the variables to the value $y_0$, the question then becomes, is the derivative of $z_0=f_y(x,y)$ defined here? $y$ is now constant, however if our derivative is defined on $f$ the partial derivative should exist.
Does it make sense to talk about the rate that $f(x,y_0)$ is changing? can we for example write $\frac{dz_0}{dx}$? or $\frac{d(f(x,y0))}{dx}$? to talk about the rate the value $f(x,y_0)$ changes? Or can we only discuss the function's partial derivatives $f_x'(x,y_0)$ and $f_y'(x,y_0)$?
Another thing is that if we apply $f$ to two arguments who depend on each other, we provide the total derivative, which is actually different depending on their relation, if the function is defined independently how can this be the total derivative of a function?
If we take $\frac{df(x^2)}{dx}$, it seems we take the 'derivative' of the value $f(x^2)$ , but perhaps this notation can be seen as the 'derivative of the function whose value is $f(x^2)$', but seems a bit strange.
Do we take the derivative of a 'value' or variable or is it better to keep Euler 'notation'.
 A: tl; dr: Functions.

In mathematics, a function is an abstraction of a deterministic relationship, often (but not necessarily) between two numerical quantities. Only once a relationship is fixed does a "rate of change" make sense. But in any case, the relationship governs the rate of change, not the quantities.
If you'll excuse a mildly provocative comment, the conceptual problem here stems from Leibniz notation, which hides functional relationships. Leibniz notation is incredibly useful when one computes, but when one is trying to understand calculus theoretically Leibniz notation (in my experience) loses out to Newtonian notation.
Let's consider the examples in question:

For example take $z=f(x,y)$, in this case we can 'fix' one of the variables to the value $y_0$, the question then becomes, is the derivative of $z_0=f_y(x,y)$ defined here? $y$ is now constant, however if our derivative is defined on $f$ the partial derivative should exist.

From a Newtonian perspective, we've defined a new function of one variable, say $g(x) = f(x, y_{0})$. The "derivative with respect to $y$" makes no sense.

Does it make sense to talk about the rate that $f(x,y_0)$ is changing? can we for example write $\frac{dz_0}{dx}$? or $\frac{d(f(x,y0))}{dx}$? to talk about the rate the value $f(x,y_0)$ changes? Or can we only discuss the function's partial derivatives $f_x'(x,y_0)$ and $f_y'(x,y_0)$?

Here, in the preceding notation, it's reasonable to interpret "the rate of change" as $g'$. That's a "total derivative" of $g$. It can also be interpreted as a partial derivative of $f$ with respect to its first variable, namely $D_{1}f(x, y_{0})$ in the notation of Spivak's Calculus on Manifolds.

Another thing is that if we apply $f$ to two arguments who depend on each other, we provide the total derivative, which is actually different depending on their relation, if the function is defined independently how can this be the total derivative of a function?

This is a reasonable question, but exemplifies why Leibniz notation is a source of confusion. Standard quasi-paradoxes with the multivariable chain rule are often set up in this framework. Writing $w = w(x, y, z) = w(x, y, z(x, y)) = w(x, y)$, for example, is asking for trouble on more levels than I care to count. These tangles can be reconciled by carefully defining functions, using different letters for different deterministic relationships.

If we take $\frac{df(x^2)}{dx}$, it seems we take the 'derivative' of the value $f(x^2)$ , but perhaps this notation can be seen as the 'derivative of the function whose value is $f(x^2)$', but seems a bit strange.

From a Newtonian perspective, we're introducing $g(x) = f(x^{2})$, so $g'(x) = 2xf'(x)$ is "the rate of change" by the chain rule. One might, I suppose, expect instead that "the rate of change" is $f'(x^{2})$, the derivative of $f$ evaluated at $x^{2}$, but I think that is not how most people would read $\frac{df(x^{2})}{dx}$.
Incidentally, Leibniz notation makes writing $f'(x^{2})$ inconvenient at best. As a result, at least some calculus students develop a tacit misconception that evaluation and differentiation commute. This bold assertion is based on experiences teaching the multivariable chain rule.
