Can a discrete valuation ring be finitely generated over a field?
You're correct: a discrete valuation ring cannot be finitely generated over $k$. Suppose $R$ was finitely generated over $k$. Then $R[t]$ would be as well, and so would $R[t]/(t\pi-1)$. But this last ring is the field of fractions of $R$, which is of transcendence degree one over $k$, contradicting Zariski's lemma (your proposition 2).