Imagine a cave in a vulcano, half way up the summit. Stalagmites and stalagtites form mostly from calcium carbonate, so I wonder whether lava has the ingredients to be dissolved by water to combine with atmospheric CO2 to precipitate as CaCO3. I know that there are many different types of lava, but does it ever include significant amounts of Ca, in other words: if I enter a cave and see stalagmites and stalactites, can I infer that the geological origin of the surroundings is not volcanic?
Well I have been in lava tubes in Hawaii that definitely had stalagtites and I think stalagmites. And they definitely weren’t calcium carbonate. 
Molten rock did drip from the ceiling when the tube was formed. They weren’t very big, as I remember them, but definitely spikes that were part of the ceiling.
So, no.
One thing that would work against them forming is that in normal limestone caves the stalactites and stalagmites form through the dissolving of calcium carbonate, and then evaporation, so they form as residual calcium crud adhering seamlessly to earlier crud.
Volcanic materials don’'t dissolve in water - even the finest particles are pieces of vapourised rock or volcanic glass which can definitely be carried along like a sediment but as particles, not solution. Once the water dissolves this will be left, but is not I think going to cement itself together without additional pressure.
Our basement garage used to have little stalactites from leaching concrete, but that’s very much a human model of a limestone cave.
Do caves naturally form in the sides of volcanoes in the first place?
Hell yeah. Lava tubes. Added emphasis mine:
As islands created by volcanoes, Hawaiʻi is criss-crossed by countless lava tubes. These underground passageways, also known as pyroducts, are created by lava flows themselves and are capable of transporting great quantities of lava long distances underneath the surface. When the supply of lava stops at the end of an eruption, or if it gets diverted elsewhere, it leaves behind an empty cave.
When a lava tube is active, lava travels along its floor at temperatures that exceed 2,000º F (1090º C). Winds of superheated fume may blast through the tunnel, yet the only sound may be the constant soft hiss of the relentless flow.
Once lava subsides, these subterranean corridors become home to unique ecosystems of troglobites, animals specifically adapted to live in this dark isolated world. Distinct species of crickets and spiders develop alongside special microbial colonies found nowhere else.
The Kazumura lava tube system, within the 500 year-old ‘Ailā‘au lava flow of Kīlauea, is more than 40 miles (65 km) long and is thought to be the longest lava tube cave in the world. Tubes may be up to several dozen feet wide.
Definitely yes.
Definitely no:
(from an inactive crater at the above volcano - sorry, image won’t preview)
(source)
Some do.
ETA: Also, even if the lava/volcano doesn’t have carbonate (doesn’t have to be calcium, BTW, can also be magnesium), it can still get carbonate deposited in it from groundwater that has traveled through surrounding carbonate rocks, say if your volcano erupted through limestone.
Those are formed by vapour deposition or hydrothermal deposition, not molten lava. But yes, they are not carbonates.
Note that you can get molten-rock drip formations as well, but the tubular ones in Hawaiʻi are not it. Even though the Wiki article talks about “lava” in reference to the liquid that forms them, it’s more properly associated superheated water-based hydrothermal fluids, not lava proper, that is the carrier for the deposited minerals.
Thank you for the comprehensive and clear answers, MrDibble, that pretty much solves my question.
Vulcanic, volcanic, vulkanisch, it may simply depend on the language Spock spoke.
I easily get confused with those vowel shifts, I hope it is not contagious like yawning.
One other thing - you asked whether lava ever includes significant amounts of Ca, but I think it’s worth pointing out a misunderstanding I think you may have - Lots of other minerals have variants that contain significant calcium, including some of the most important silicate minerals that make up the most common lavas:
Plagioclase feldspars, pyroxenes and amphiboles.
So in a way, almost all lavas contains Ca.
But these are silicates, and so the calcium is bound in the rock until the mineralogy is altered, by either weathering or metamorphism.
I knew it would not be easy or simple! Good to know that not everything that contains Ca is a carbonate but that it can turn into one by weathering or metamorphism. That makes a lot of sense. Thanks again!
I suppose that, if you get any kind of rock hot enough, it’ll melt. And the molten rock will still have the same composition as the original, as it will when it re-solidifies (though possibly in different compounds, or with a different structure). So it’d have to be possible to have carboniferous igneous rock.
Not exactly - stuff can escape as gases. Especially CO2
I’ve already cited where it’s more than possible - BUT carbonatites are shown by isotope analysis to not come from molten sedimentary carbonates, but instead from partial melting and fractional crystallization of alkaline magmas.
When sedimentary carbonates get heated, they release their carbon as CO2 way before they get to melting temperatures.
Ah. That had occurred to me, but I rejected it on the grounds that this could all be happening far from atmospheric oxygen. I’d forgotten that the rocks themselves would contain oxygen, too.
If you’re ever in Bend Oregon, I recommend a visit to Lava River Cave (don’t forget your flashlights!)
From the Forest Service website A Journey Through Lava River Cave
In other areas of the tube volcanic stalactites occur. These “lavacicles” are found in two forms: hollow cylindrically shaped “straws,” formed by escaping gases, and the cone-shaped drip pendants, formed by remelted lava dripping from ceilings and walls.