Does the Moon's core still contain significant heat? On earth, using earth-sheltering techniques can significantly reduce the temperature fluctuations on a structure. Would the same statement be true as well on the Moon?  Does the Moon's core still contain significant heat?
 A: The core of the Moon has solidified.  This is evident from the fact that the Moon always show the same side to the Earth (except for a bit of wobbling).  Consider eggs - hardboiled eggs are much easier to make rotate by spinning than raw eggs.
How much it has cooled below the melting point I do not know, and whether you consider this significant.
A: No, the moon formed when a small mars-sized planet crashed into the proto-earth. This was after the earth was almost fully formed so all the denser materials had sunk to the core and the outer ones "fell off" and became asteroids. These less dense asteroids flew off and were slowly pulled together to form the moon. Therefore all the hot parts of the earth had cooled by the time the moon formed. This means the core may still have some heat but not a significant amount.
A: On the earth-sheltering question, the answer is yes, using material to increase the thermal mass of structures would work just as well on the Moon as on Earth.  There might be minor differences due to different materials and lack of water in Moon soils but the general principal would still apply.
As for the Moon's core still containing significant heat, that answer is no.  At least not compared to the Earth.  Smaller bodies cool much more rapidly than larger bodies as their surface area to volume ratio is much higher and therefore they can radiate heat faster.  The Moon's core cooled off much, much faster than the Earth's and most of the latent heat of formation is now gone.  There is probably still some heat left from decay of radioactive materials but it is much lower than what the Earth has.  
An indicator of the lack of interior heat is that the Moon is very seismically quiet, there isn't much going on under the surface.  Seismographs place by the Apollo missions showed very, very little activity.  If there were heat and significant liquid portions to the Moon's interior, there would have been much more activity on the seismographs.
A: It depends on, what consider we significant. On our current best hypothesis, the moon was created by a collision of the Earth and a Mars-sized planet. Thus, the core of the Moon contains much fewer heavy elements, and thus also much fewer Uranium and Thorium. This corresponds also the fact, that themean density of the Moon is only 3.3, while the Earths is 5.5.
But the few uranium and thorium, whose radioactive decay heats the planets, is still there and their half life is still as long, as in the Earth. It won't be slower only because the Moon is smaller, although the heat loss by thermal conduction is relative faster because its smaller size.
On the current best estimation, the temperature in the core of the Moon is around 1600-1700K, although its core is significantly smaller, so there is no plate tectonics or any other, so getting geotermal energy with technologies similar to the current Earth, would be probably highly infeasible (it would require thousand mile deep wells).
A: The core of the Moon must be far hotter than the maximum temperature reached by its surface.  This is because gravity maintains a temperature gradient in solids, liquids and gases.  The gradient represents a state of thermodynamic equilibrium with maximum accessible entropy, as the Second Law of Thermodynamics says will eventuate.
