How vertically deep do lava conduits go?
As mentioned in the comments, lave caves and tubes are surface features. You get a lava tube when there's a lava flow and the external part in contact with air (or water) cools down and solidifies while the interior is still flowing. National Geographic made an excellent feature on it, all with diagrams and photos and stuff.

Why do these tubes not go far deeper?

Because lavas are liquid, and they flow downwards. If there's a hole that goes deep, the lava will fill it. It's possible to explosively eject lava out of a hole (as commonly happens), but these are violent events that will cause the collapse of the conduits, forming breccia pipes, with collapsed calderas.

I think of magma - liquefied hot rock erupting out of volcanoes - as coming from even deeper than that depth
...terminating at the threshold of the asthenosphere?

Let me clear a common misconception here: there is no magma ocean in the interior of the Earth. There is no large reservoir of magma in the deep earth, and volcanoes are not just holes through which this (non existent) magma is coming out. Both the crust and the mantle are essentially solid rock. The asthenosphere is solid rock.
Melting is an uncommon occurrence in the Earth's crust and mantle. It needs exceptional circumstances to happen, which mostly happen above subduction zones (i.e. the Pacific Ring of Fire) or above hotspots (East Africa, Hawaii, Yellowstone). The erupting magma can come from anywhere that's as shallow as less than a kilometre below the surface (which can be above sealevel if you're on a high volcano) to several tens of kilometres. Very rarely these magmas come directly from asthenospheric depths. These magma chambers can have complex plumbing systems beneath them and all sorts of morphologies of feeders.