Let $f:Y\to X$ be a proper birational map of normal varieties over an algebraically closed field which is an isomorphism over the regular locus.

Q1: Is it the case that the pullback $f^*\operatorname{Pic}(X)\to\operatorname{Pic}(Y)$ is always injective?

(Update: this was answered in the affirmative by Jason Starr.)

Q2: Suppose that there exists a finite set $\{C_i \mid i\in I\}$ of projective curves in $Y$ such that a line bundle is ample (respectively nef) on $Y$ if and only if its restriction to each $C_i$ has positive (respectively non-negative) degree. Is it the case that a line bundle on $X$ (interpreted via pullback as a line bundle on $Y$) is ample (respectively nef) if and only if its restriction to each $C_i$ that isn't contracted by $f$ has positive (respectively non-negative) degree?

For example, suppose that $Y$ is a crepant resolution of a Kleinian singularity, with exceptional divisor $E\subset Y$ consisting of a finite union of projective lines. Suppose that $X$ is a partial resolution, obtained from $Y$ by contracting some (but not all) of the components of $E$. Then the Picard group of $X$ should be isomorphic to the subgroup of the Picard group of $Y$ consisting of line bundles whose restrictions to the contracted curves have degree zero, and its ample cone should be those line bundles whose restrictions to the un-contracted curves have positive degree.