Can we choose values for multivariable functions deliberately?

This question is no "silly" at all! I would consider it rather "to-the-point" of multivariable calculus.

In general, as you mentioned, the set $$G(f)=\{(x,y,z)\in\mathbb{R}^3|z=f(x,y)\}$$ is indeed a subset of $\mathbb{R}^3$ and, in general, a surface. So, you can pick arbitrarily coordintates $x,y$ in $f$'s domain. Or, to take it to the multidimensional level, you can pick elements of $\mathbb{R}^2$ of the form $(x,y)$ that do belong in $f$'s domain.

Now, as for the circle, well, $x^2+y^2=r^2$ is not a function with free variables $x,y$ - you noticed they are not free. However, if you consider the function $g(x,y)=x^2+y^2$ this is a function of $x,y$ defined on $\mathbb{R}^2$.

Now, if the two coordinates have some kind of dependence from one-another, the thing gets more compicated. Imagine the situation where $$y=3x+2$$ In that case, it is clear that: $$f(x,y)=f(x,3x+2)=h(x)$$ for some other, one-variable, function $h$. In our case, if $f(x)=x^2+y$, then $h(x)=x^2+3x+2$. You can imagine this dependence between $y,x$ as a choice of a "path" on the surface that $f$ defines.

Lastly, in many cases, I see many-dimensional situations - since, well, for $n\geq4$, $\mathbb{R}^n$ is a little "scary" place - as a degree of freedom. With two free variables I have the ability to move over a surface, whereas with one free variable, I have the ability to move over a curve etc.

In general $$f(x,y): D \subseteq \mathbb{R^2}\to\mathbb{R}$$is defined in a subset $D$ of $\mathbb{R^2}$ that is the real plane.

In your case $$f(x,y)=x^2+y$$

is defined for each value of (x,y) thus $D\equiv \mathbb{R^2}$.