Sorry! It was rather easy, so perhaps it should be closed.

The second question clearly gets a negative. Let $\mu$ be a finitely additive measure on $\mathbb Z^+$ that assigns zero to all finite subsets. Let $n=1$. Let $f(k)=1/k$. Then $\int_{\mathbb Z^+} fd\mu =0$, but $0$ is not in the convex hull of the range.

The first question gets a positive. It's clear for $n=1$. Suppose $n>1$ and it's true for smaller dimensions. Let $e=\int_\Omega fd\mu$. Let $C$ be the convex hull of the range of $f$. If $e\notin C$, there is a hyperplane $H$ through $e$ such that $C$ lies to one side of it. Let $p$ be a normal of $H$ such that $p\cdot f(x) \le p\cdot e$ for all $x\in\Omega$. Then $\int_\Omega (p\cdot f) d\mu = p\cdot e$ but $p\cdot f(x) \le p\cdot e$ for all $x$, so $p\cdot f = p\cdot e$ almost everywhere. We can modify $f$ without changing its range so as to ensure $p\cdot f = p\cdot e$ everywhere. But then the range of $f$ lies in the $(n-1)$-dimensional hyperplane $H$, and so by the $(n-1)$-dimensional case we have $e$ in the convex hull of the range of $f$.