Re: "shut down"
some of that depends on the design. Usually in an emergency they would drop all control rods to the bottom and then inject a boron-rich "poison" into the coolant (typically this is part of how power levels are controlled anyway), sufficient to keep the thing from re-starting under any circumstance. Then, since civilian reactors (with lower enrichment) typicall have positive temp coefficients, as long as decay heat is removed criticality will never be reached (or if a lack of heat removal melts the core, for that matter). This can typically be done by bleeding steam if nothing else is available. For "reasons" if worse things happen (like overpressure, uncovered core, etc.) the physics would (or maybe 'should') not support criticality.
So once the reactor is poisoned and any emergency cooling systems have been enabled, it would remain shut down. It could melt, but it wouldn't restart (unless the design really really sucked).
Fukushima (in part) was similar to the worst case scenario. The reactors shut down but decay heat apparently caused meltdowns and Zr-water reactions (which produces hydrogen gas bubbles inside the core and whatnot), but as far as I know they did not regain criticality. The emergency shutdowns apparently prevented an uncontrolled restart but the lack of electricity did nothing to remove decay heat. And so, "it happened".
Still, horrible disaster averted in Ukraine.