> The core contains 1e31 J of energy

What does that mean? Does it mean the core temp will drop to zero degree kelvin when 1e31 J is removed from it?

Can we realistically bring the core temp down to absolute zero? I think you need a temperature difference between a "source" (here earth's core) and a "sink" (here surface of the earth (radiation), atmosphere (convection)) to drive any engine. So geothermal energy stops being useful when this difference is zero or less?