Can lava that has cooled to a hardened state be returned to a molten state by, say, a new river of molten lava flowing over it - like re-smelting iron? If you threw a lava rock into the mouth of a volcano, would it remain in its solid state or would it melt and become part of the cauldron of magma?
Yes, at least partially.
It is important to remember that a "rock" usually does not have a single melting point. It melts over an interval. Igneous (magmatic) rocks are made of combination of minerals (for example, quartz, olivine, pyroxene, feldspar) that each will melt at a different point. When a rock is heated up, the minerals with the lower melting point melt first, followed by the higher melting point minerals with progressive heating.
Some rocks are composed mostly of easier-to-melt material (for example the stuff that came out of Mt. St Helens, mostly granitic composition) and others are made of harder-to-melt material (for example Hawaiian basalts). Taking a a granite and throwing it into a basalt will melt most of it for sure, if not all. Not the opposite though.
Even if you do not completely melt the rock, you will disintegrate it enough that the remaining harder-to-melt minerals will just drift away and mix with the lava, making it impossible to distinguish the original rock any more.
Likewise, when hot lava flows on easy-to-melt material (for example clay-rich soil), some of it will melt or vitrify (turn to glass).
I will finish by emphasising that all lavas, every single one of them, were originally solid rocks that melted at depth by some kind of process. So yes, it is possible to melt rocks.