This question concerns a previous question, Rotation number of inverse maps on the circle. in which all the terminology and notation used below is defined.
The question is given the rotation number $\rho (f)$, calculate the rotation number $\rho (f^{-1})$. All proofs I have found of this statement calculate $\rho(F \circ F^{-1})$ (for $F, F^{-1}$ lifts of $f, f^{-1}$) and end up with something like:
$\rho(F\circ G)=\lim_{n\to\infty}\dfrac{(F\circ G)^n(x)-x}{n}= \lim_{n\to\infty}\left(\dfrac{F^n(G^n(x))-G^n(x)}{n}+\dfrac{G^n(x)-x}{n}\right)=\rho(F)+\rho(G)$
For $F, G$ lifts of commuting circle homeomorphisms $f, g$, and the proof ends by letting $G = F^{-1}$. But in the definition of rotation number for $f$, i.e. $\rho(F) = \lim_{n\to\infty}\dfrac{F^n(x)-x}{n} = \lim_{n\to\infty}\dfrac{F^n(x)}{n}$, if we let $x = F^{-n}(y)$ for some $y$ as above we get that the rotation number is $\rho(F) = \lim_{n\to\infty}\dfrac{y}{n} = 0$ for any $f$ (note that the rotation number is always defined and has the same value for any $x$ in the domain of $F$).
So my question is are we allowed to use a function $G^n(x)$ in place of a variable $x$ in the definition of rotation number as the proofs commonly do, or are these proofs incorrect?
You are pretty confused. They are used as such in the definition. When you substitute $x = F^{n}(y)$, it changes to $y - F^{-n}y$. Since $f$ and $g$ commutes, their lifts too. Only that property is being exploited here.