A quick note that (2) doesn't follow from ZF (assuming ZF is consistent) or even ZF+DC.
(2)+Countable Choice for Finite Sets (CCFS) implies that every uncountable set has a non-principal (finitely additive, non-trivial) measure. Let $\Omega$ be uncountable. Let $\mu$ be as in (2). Let $A = \{ x : \mu(\{ x \}) > 0 \}$. The standard proof that a finite measure has only countable many atoms only uses CCFS (there are at most $n$ elements $x$ such that $\mu(\{ x \}) \in [1/n, 1/(n-1))$). So $A$ is countable. So $\mu(A)<1$. Then $\nu(B) = \mu(B-A)$ defines a non-principal measure and is non-trivial since $\nu(\Omega)=1-\mu(A)>0$.
But Blass gave a model of ZF+DC with no non-principal measures ("A model without ultrafilters", Bull. Acad. Polon. Sci. 25 (1977), 329–331; I haven't read the paper, but Pincus and Solovay say that it proves this). So (2) is false in that model.