Criteria for $f(f(x))=g(x)$

Question

I 'm searching about the solvability of the functional equation $f(f(x))=g(x)$. I have three questions about it:

1. Let's be $g$ an arbitrary function and the functional equation $f(f(x))=g(x)$. Are there any specific criteria to ensure us that there exist such function $f$?
2. Let's be $g\in C^0$ and the functional equation $f(f(x))=g(x)$. Are there any specific criteria to ensure us that there exist such function $f$ and $f\in C^0$?
3. Let's be $g\in C^1$ and the functional equation $f(f(x))=g(x)$. Are there any specific criteria to ensure us that there exist such function $f$ and $f\in C^1$?

Thanks in advance!

P.S.: I read this and this, but there are a little bit different questions.