Does the athenosphere produce surface lava?
The chemistry of volcanic rocks gives a lot of clues as to the origin of the melt. The athenosphere is actually a bit poorly defined in this sense (it's a mechanical classification, not a chemical classification). Regardless, it's easy to distinguish magmas with a pure mantle source from others. MORB (mid ocean ridge basalt) is a common acronym for volcanic rocks with a chemistry that indicates they're essentially pure mantle melt.
In the most common type of volcanoes you see on land (arc volcanoes), magma forms due to the introduction of water and other volatiles that lower the melting point of the upper mantle. That's what happens at arc volcanoes like the Cascades or Mt Fuji in Japan. It's not exactly coming from the athenosphere in that case, and it's often the lithospheric mantle and lower crust that are being melted. It's not heat that causes it, but instead the introduction of water.
Volcanoes can also form due to the introduction of extra heat, as happens above hotspot volcanoes (e.g. the Galapagos or Hawaii). In many of those cases, you're basically seeing the athenosphere supplying extra heat to the mantle lithosphere and crust and melting it. The magma usually isn't coming directly from the athenosphere in those cases (though some of it can and does). However, flood basalts are an extreme case of hotspots, and their chemical signatures suggest that it's more or less melt directly from the athenosphere.
Finally, one of the most common types of volcano on the planet does come directly from the athenosphere: mid ocean ridge systems. (Where the term MORB comes from) Those are almost entirely deep in the oceans, so non-geologists don't think about them as much, but they make up the bulk of volcanic activity on Earth. In those cases, you're essentially bringing the athenosphere up and melting more and more of it as it rises. Those are the cases where magma is most directly sourced from the athenosphere.
Where does the water come from? Is it liquid, or gaseous?
As for whether it's liquid or gaseous, I probably shouldn't have called it "water", and should have said "hydrogen and oxygen in silicate minerals". Silicates always have oxygen, but some hydrogen too (hydroxyl groups - i.e. OH).
Basically, you have minerals that transform to other minerals at depth due to the pressures and temperatures involve. That transformation leads to the release of hydrogen and oxygen as they're in one crystal structure and not in the other. (Basically, minerals with hydroxyl groups transform into minerals that do not have hydrogen in their composition, releasing hydrogen and oxygen in the process.) That hydrogen and oxygen diffuses into adjacent mineral crystals and can cause them to melt.
With that said, any time you have magma (i.e. melt), it's going to have H2O, CO2, halogens, etc dissolved in it, just like water has oxygen and carbon dioxide dissolved in it. We talk about these in simple terms of "H2O" and "CO2" etc, but often the details of how things are bonded are a bit different, just like CO2 dissolved in water isn't exactly CO2, but is carbonic acid (H2CO3) instead.
When sugar dissolves in water, it creates a solution, a chemical mixture characterized in this case by ionic bonds between water molecules and sucrose molecules. In general, the melting and boiling points of a solution will not be the same as the melting and boiling points of any of the individual substances that comprise the solution.
When I think about combining substances, I think about suspensions, solutions, colloids. (When I think about combining substances, I'm usually thinking about cooking). A suspension is small particles mixed into a liquid, but it isn't stable; after time or with a centrifuge, you can separate them back out. A solution is a chemical bond between individual particles of each substance, such that it is stable. A colloid is small particles mixed into a liquid or gas, but the small particles are so small that it is stable and cannot easily be separated out. Here's the link I used to refresh my memory enough to explain these concepts: https://lab-training.com/understanding-differences-solutions...
I'm sure you're aware of all that, but I wanted to write it out so I could be more explicit about my confusion. Adding water to solid rock will not result in a solution, colloid, or suspension. Adding finely ground rock and water can do so; that's kind of what mud or clay is. So I was confused how adding water to rock could result in any kind of mixture that changes the melting point of the rock.
Jofer's explanation makes sense to me: it's not actually adding H2O molecules to solid rock. Instead, it's that certain minerals under sufficient heat and pressure change mineral/crystal structure in a way that results in the precipitation/expulsion of various molecules and ions, including hydrogen and oxygen. Those precipitates diffuse through the solid until they react and combine with other minerals, causing transformations of those into new mineral/crystal structures with a lower melting point.
The link you provided talked about how solutions of sucrose and water in differing concentrations have different melting points, but I was confused about how rock and water could dissolve at all.
The general thing that is happening is this: water allows the sugar molecules to separate without paying as large an energy penalty. This means the higher entropy of disordered sugar molecules allows that energy penalty to be overcome at lower temperature (and/or higher concentration, which reduces the entropy increase.)
The same thing is happening with water and rock, if not by exactly the same mechanism. Water reduces the energy cost of breaking bonds in the rock (covalent Si-O bonds, not hydrogen bonds).
Adding water to rocks does result in a solution. Rock is very slightly soluble in water. It's just that at normal temperature and pressure the solution is extremely dilute. This is because the energy penalty is so high that a very large entropy increase is needed to overcome it, and the entropy is logarithmic in dilution. Raise the temperature a bit and the solution becomes more concentrated. This is how quartz is deposited to grow artificial crystals for electronics: it goes into solution in hot, pressurized water and is deposited on seed crystals. This also happens naturally with various minerals (including quartz.)