Question: Is it consistent with ZFC that every pair of disjoint sets $A,B\subseteq\mathbb{R}$, both of size $\aleph_1$, can be separated by a Borel set?
This statement is clearly false under CH; take any non-Borel set $A\subseteq\mathbb{R}$. Then $|A| = |\mathbb{R}\setminus A| = \aleph_1$, but $A$ cannot be separated from its complement by any Borel set. My guess is that it's always false, i.e. the answer to the above question is no. A Hausdorff gap would seem to be a good way to get a counterexample, using Todorcevic's theorem on analytic gaps (see here), but I can't seem to get the right statement to come out. Maybe there's an easier counterexample?
Yes, for example, if every set of reals of size $\omega_1$ is a Q-set (this means that every subset is relatively $G_{\delta}$), then you can separate them by a $G_{\delta}$.