What type of temperatures, and what type of torch can one use to melt rock?
The short answer is that rocks are made out of different minerals and each of those minerals have their own melting temperature. When you melt a rock, the whole thing doesn’t just melt in one piece (unless it’s very homogeneous)-- the low temperature minerals melt sooner so you’d end up with a sort of magma slushie. So the answer to what temperature is required is a resounding “it depends”. For a very ball park figure, the minerals commonly found in silicate rocks are all solid below 600 C and all liquid at 1200 C. The exact details of where you get first melt and total melt and what happens in between is the stuff of the phase diagrams that fill up petrology books.
If you’re talking about melting rocks so they’re 100% liquid and runny, you generally need an industrial/laboratory furnace. Some pottery kilns can melt certain rocks too. I’ve seen some pretty spiffy pottery that took advantage of this, but I guess it usually happens when some hapless art student decides to bake in some cool rock they found at the beach. You can also partially melt some rocks in a particularly hot campfires, although don’t expect to see anything too exciting-- some granites will crumble as the low temperature minerals melt and you’ll end up with a pile of mineral grains in the morning. Usually a bunch of partially melted beer cans too.
“Melting rock” and “creating artificial igneous rock” are not the same thing. The *only *igneous rock you’ll get with any torch is some form of glass (or aphanite if you’re really lucky and careful with the cooling). To create actual artificial rock, you’ll need a furnace that is capable of some pretty precise control and high temperatures in the 600-1400 degree region, to get decent crystal formation. If you want to replicate most of the funkier rocks, you’ll also need a way to add and extract material and volatiles during the process. Furnaces like that exist at some experimental geology labs, but I’ve never been lucky enough to work with one. Most petrology melting experiments are done on glasses.