Do black holes have measurable mass or energy?
We can measure its gravity, so certainly.

When they swallow stars and galaxies do they not break the first law of thermodynamics for the closed system of the universe?
This is more complex and I should probably leave this to mojo. But I won't.

I don't think so. Because it has gravity, its mass still exists in this universe, it remains part of the closed system that is the universe. It hasn't gone anywhere. The total mass/energy of the universe remains unchanged.

The question for me really is... does a black hole occupy space?

If not, does it travel through space? Or does space travel around it? Is there even a distiction between those two concepts?