Melting basalt on the lunar surface - what would the resulting material be like?
Melting lunar dust of basaltic composition will result (unsurprisingly) in basalt lava. There might be some volatile loss - if for some reason the basalt had some H2O or CO2 trapped in the solid you will most likely lose it. It's probably not going to change its chemical properties much though.

On Earth, as lava cools down fast, for example when it's erupted underwater, or when it's the top of a lava flow in contact with air, it quenches to volcanic glass. When it cools down slower, there is enough time for small crystalline material to form in it. These are usually tiny crystals in the μm size. On the Moon, I reckon the lava would cool down slightly slower than on Earth (since there is no air to conduct away heat), but since you're talking about a lava layer which is only 2 cm thick I'm pretty sure it will cool down fast enough for the formation of glass, and any crystalline portion of it will be negligible. I am not sure whether it will crack or not. From my own experience of melting rocks, when you have a similar amount cooling down in Earth's atmosphere, they crack. But then, silicate glass does not crack when cooled down in air. I guess it comes down to the composition of the rock/glass.

The lava will not percolate down to the regolith. Lavas hardly penetrate soils on Earth, so it is probably not going to happen on the Moon where the gravity is much weaker.

If I were looking for construction materials on the Moon, I would just build a quarry. Seems easier than melting down dust.