Are there Black Holes that do not have global time-like killing vector fields? Can there be any Black Holes that do not have global timelike Killing vectors? And if there can be such Black Holes, does the conservation of energy apply to them? Can there be Black Holes with a non-asymptotically flat spacetime, so that conservation of energy doesn't apply well to them?
 A: Your layout of the issue in your question is not right.

Energy is only conserved in particular kinds of reference frames, and you can't cover the universe in one.

The condition for conservation of energy doesn't have anything to do with frames of reference, which are not really a thing in GR. There is a Komar mass that is conserved for stationary spacetimes, and an ADM mass that is conserved for asymptotically flat spacetimes. We don't have a conserved energy in cosmological spacetimes because (like almost all spacetimes) they're not stationary or asymptotically flat.

The precise mathematical way to formulate energy conservation in general relativity is through timelike Killing vector fields,

No, this is something much more specialized. This is the question of whether you can define a conserved mass-energy for a test-particle.
Black hole spacetimes are asymptotically flat, so they have a conserved mass (the ADM mass). This is why we can talk about "the" mass of a black hole. There is nothing super mysterious about this mass. It's simply the mass you'll detect if you measure gravitational forces at large distances and apply Newton's law of gravity. For instance, this is how we know the mass of Sag A* -- from the orbits of the stars orbiting it.
