Looked up some papers, and seemingly super low compared to what I would have initially guessed - probably because it’s porous/fluffy/sharp dust with lots of small voids/less compacted I’m guessing. Like, orders of magnitude less than the ground on earth. Not my area of expertise though and was just cursorily skimming papers for values. Specific heat cap and density seem like what you would expect for any rocky materials.

Might need to pour a bunch of water on the soil to increase the thermanl conductivity and make sure to never heat it above freezing?

Water would boil on the moon, any ice would sublimate. You would be better off just compacting the regolith.

Geothermal heat transfer is greatly affected by the moisture of the soil, which on the moon would be pretty low as you can imagine.

No one ever drilled the Moon to any useful extent. All we know is that it's mostly dusty on top.