Does there exist a non-measurable function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f ^{−1} (y) $is measurable for any $y \in \mathbb{R}$? Does there exist a non-measurable function 
$f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f ^{−1} (y) $is measurable for any $y \in \mathbb{R}$?

Let $A \subseteq \mathbb{R}$ be a non-measurable set such that $|A| = |\mathbb{R} \setminus A| = \mathfrak{c}$. 
Let $f : A \to \{x \in \mathbb{R} \mid x < 0\}$ and $g : \mathbb{R} \setminus A \to \{x \in \mathbb{R} \mid x \geq 0\}$ be bijections. Then $(f \cup g) : \mathbb{R} \to \mathbb{R}$ is a non-measurable bijection (as the preimage of the open set $\{x \in \mathbb{R} \mid x < 0\}$ is $A$) . As the preimage of each point is a singleton, your condition is satisfied.
