Let $f: Y \to X$ be a birational morphism of projective varieties. Let $\mathcal{M}$ be a very ample invertible sheaf on $Y$. Suppose also that:

$f^{-1}$ is defined away from a single point $x \in X$.

$f_* \mathcal{O}_Y = \mathcal{O}_X$.

Two questions:

(1) If $V$ is a set of global sections of $\mathcal{M}$ that generate $\mathcal{M}$, consider the induced evaluation map $ V \otimes \mathcal{O}_X \to f_* \mathcal{M}$.

Let $\mathcal{N}$ be the image of this map. Is it possible for $\mathcal{N}$ to be reflexive? (Full disclosure: I would like the answer to be no.)

If we know that $X$ is smooth at $x$, or more generally that $(f_* \mathcal{M})^{\vee\vee}$ is invertible, then the answer is no. Let $E$ be the exceptional locus of $f$. Then $E$ is positive-dimensional, so any hyperplane section meets $E$ nontrivially, and thus any section of $f_* \mathcal{M}$ must vanish at $x$. But if $(f_* \mathcal{M})^{\vee\vee}$ is only reflexive, then I don't see how to generalise this argument.

(2) Is it possible for $f_* \mathcal{M}$ to be reflexive? If so, what is the weakest possible condition that will guarantee that $f_* \mathcal{M}$ is not reflexive?

My chief interest in (2) is that a negative answer to (2) would be a cheap way to get a negative answer to (1).