A wave is a phenomenon that propagates through a field - i.e. the field is what allows the wave to exist.

(The philosophy of that admittedly gets messy, though, e.g. "are fields real objects?")

Yes, very messy and ultimately unknowable.

Sure. We can say the same thing about a "quantum wave", though.

Yes. This is why Physicists will reject the "everything is a wave theory" till the bitter end. They become frustrated when faced with the un-measurable.

There is no "quantum wave", there are only waves. Immeasurable, undefinable waves.