Considering lava, it seems like you might say that the majority of the material inside the earth is molten today.
No, due to the pressure within the mantle and even between the crust and the mantle. The asthenosphere, for example, is generally solid due to that pressure; it’s actually when the pressure goes down that molten magma can form, as I understand it.
Also, the planet’s outer core is liquid iron and nickel, but the inner core is solid; but the temperatures down there are as hot as the sun’s surface.
One thing that magma depends on to form is dissolved gases, particularly sulfur compounds, carbon dioxide and even water. Another is decompression in the asthenosphere, a layer of the upper mantle that is solid yet plastic.
Phenomena like this usually happen at fault lines; those are not conduits for a giant molten mass underneath the ground to pass through, but areas where compression and decompression happen for these phenomena to occur, otherwise we could have volcanoes everywhere.