Yes. Suppose $f$ contracts a curve $C$. Then $K_{\tilde V}.C<0$. But $K_{\tilde V}=f^*K_{V}$ by your hypotheses on the exceptional locus. And so $K_{\tilde V}.C=f^*K_V.C=0$, a contradiction.

Yes. Suppose $f$ contracts a curve $C$. Then for any ample divisor $D$, we have $D\cdot C>0$. But $D=f^*f_*D$ by your hypotheses on the exceptional locus, and so $D\cdot C=f_*D\cdot f_*C=0$, a contradiction.