When is a local, reduced, (commutative) ring an integral domain?

This is not true. The ring $k[x,y]/(xy)$ is reduced but the localization at $(x,y)$ is not an integral domain. For instance, because $xy = 0$ in the localization but $x \neq 0$ and $y \neq 0$.

There is not a whole lot to say here. You are given that $\{0\}$ is semiprime, and you are asking when it is prime. You could say that $R$ is a domain iff $\{0\}$ is primary. fpqc's example demonstrates this nicely since $(xy)$ is a semiprime but not prime ideal of $k[x,y]$ and also not in the localization at $(x)$ or $(y)$.

Locality never really comes into play.

The reason you did not find any counterexamples using $\Bbb Z/(n)$ is that this ring is only reduced when $n$ is squarefree, and for each such $n$ the ring is von Neumann regular, and the localizations at prime ideals are all fields.