Taylor's theorem for holomorphic functions tells you that $P_nf(z)$, the $n$th degree Taylor polynomial for $f$ about $z_0$ with radius of convergence $R$, is such that $|P_nf(z) - f(z)| \rightarrow 0$ uniformly for every $z$ in a disc $D(z_0,r)$ with $r. This satisfies the criterion for convergence that you gave above, and hence for any contour $\Gamma$ in this region of convergence,
$ \lim_{n \rightarrow \infty} \int_{\Gamma} P_nf(z) \, dz = \int_{\Gamma} \lim_{n \rightarrow \infty} P_nf(z) \, dz = \int_{\Gamma} f(z) \, dz$
As each integral $\int P_nf(z) \, dz$ is just the integral of a polynomial, then you can integrate it termwise as usual. This justifies the fact that you can integrate holomorphic functions termwise.
Termwise differentiation can be obtained from the result that $f'(z)$ is itself holomorphic, and then writing it as another Taylor series.