Residue of $g(z)g'(z)$ 
I know how to use the residue theorem and the winding number to find the residue of $f$, but I have no idea how to relate the residue of $g$ to that of $g\cdot g'$, especially without knowing anything about the order of the poles. 

Hint: We can apply a substitution write
$$
\int_{\gamma} g(z)g'(z)\,dz= 
\int_{g(\gamma)} u\,du
$$
Of course, the function $\phi(u) = u$ is entire.
