It has been pointed out in the comments that Ewan's insightful solution shows that a negative answer to the question is consistent with ZF, since a positive answer implies AC. 

But let me go one further. In fact, Ewan's solution shows that a negative answer to the main question is consistent with full ZFC. The reason is that a positive answer to Ewan's problem 2 actually implies the set-theoretic assertion $V=HOD$, that the universe consists of the hereditarily-ordinal-definable sets. To see this, supoose that $V\neq HOD$, then there is some cardinal $\kappa$ such that $Y=P(\kappa)$ has some non-ordinal definable elements. Let $Y'$ be the set of non-HOD subsets of $\kappa$. Both $Y$ and $Y'$ are ordinal definable, but $Y'$ has no ordinal definable elements. This contradicts any positive solution to problem 2. 

Thus, any model of set theory having a positive answer to problem 2 must also satisfy $V=HOD$. And so any model of $ZFC+V\neq HOD$ is a model of a negative answer to the main question, with full ZFC.