First get a source of rocks, then have a smelter that you could melt rock in the same way as we melt metal every day. You would need a good bellows, and a crucible, and tongs, and enough fuel to melt the amount of rock needed. Then you would pour this new molten rock over a mound of dirt for a dome house. Or you would pour the molten rock into trenches in the ground in the shape of walls, to make a traditionally shaped house, digging out the dirt later. Hopefully all of the different pourings of liquid rock will all stick together.
The total cost of this type of house would be mostly fuel costs and labor costs. But with the cost of energy so high these days it may be prohibitive to make this type of house, seeing as how the only way we know to heat materials to such temperatures is by using coal or coke.
This is possible, but the main reason that nobody has done this yet may just be because it’s cheaper to make concrete, and cinder blocks, and brick, than to melt rock into custom structures. But cutting out the middle man may make it more cost effective. If you built it yourself and only paid for the cost of coal for the forge.
This might seem crazy, but think of the advantages if it worked. Once it’s built it will last forever. And if you cover the house with dirt then the thermal mass will keep the temperature stable year round, using the same theory as cave systems that are the same temperature year round. Also the house will never burn down, and is water proof, and is eco-friendly from less heating and cooling costs, and it’s eco-friendly to dispose of if you ever decide to tear it down. But maybe the energy costs are too great, and also they would never allow you to build it in the middle of a city from the obvious dangers of molten rock. But it’s fun to think about anyway, if only by virtue of the fact that it’s theoretically possible.
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Assuming that there's a practical way to do this I imagine not any rock would do as different rocks have different characteristics and the energy released would be phenomenal if it ever became practical and popular and do significant things to global warming.
I think this is a really interesting idea, but in the end it doesn't turn out to be that much different than a concrete house. I expect you'd still use similar forms and methods of smoothing, but you'll have to take extra care to deal with the heat. Most concrete forms are aluminum, which has a low melting point, but does conduct heat away pretty well. I don't know how it would do with lava. Floats probably wouldn't work to smooth--smoothing would probably have to be done with grinding after it cooled.
All in all, I like the idea, but I think a concrete house is far easier.