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The answer to your first question is, "no." Let $X$ be the Riemann sphere $\mathbb{C}P^1$ with the meromorphic differential $\omega = dz/z = -dw/w$, where $w=1/z$. Let $f$ be the meromorphic function $f(z) = z/(z-z_0)$ for some $z_0 \neq 0$. The only pole of $f$ is at $z_0$, and the residue of $f\omega$ at that pole, i.e., the residue of $dz/(z-z_0)$ at $z_0$, is nonzero.