bonus

This does not imply anything for f(0).
Indeed, if f(z) was replaced by ( f(z) + C ) for any complex constant C, then on
one side f ( 0 ) could be made to have any value, while on the other side f '(0 ) = 5 i

and the integral remain completely unaffected:
Note that in the integral
Int(f(z)/(z-0)^2,z)=
int(subs(z=zt,f(z)/(z-0)^2)*dzt,trange);
one of the above are affected if any constant f

> Int((f(z)+C)/(z-0)^2,z)=Int(f(z)/(z-0)^2,z);

since

> Int(C/z^2,z)=0;

as is easily verified e.g. by direct computation

> zt:=r*exp(I*t);
Int(subs(z=zt,C/z^2)*diff(zt,t),t=0..2*Pi)=
int(subs(z=zt,C/z^2)*diff(zt,t),t=0..2*Pi);

>