You are right to be confused about QM and the wavefunction. Now we have experiments, and the data from the experiments tell us what the real world looks like. We can call this the fabric of space time or fields. We do not really know what they are or how they would look or what they are made of, or what are these waves made of.
What we do know is the data from the experiments.Then we try to build mathematical models to try to model the real world. As of today, the model that describes the data best is the standard model and QM.
Now you are confused whether QM and the wavefunction is the real world. QM is a theory, a mathematical model that best fits the data. We use the wavefunction to describe the probability destribution of the elementary particles in all of space.
You are asking about QFT, where each elementary particle is represented by a field (and its excitation). You are asking too, whether the excitations of these fields, the elementary particles, are propagating as waves. Now again, you confuse the real world with the mathematical model. There is a real world, where elementary particles propagate through spacetime, we know that from the experiments. Now the model again that best fits the data would be QFT and in the math, you could represent the propagation of these elementary particles as waves. The reason why we represent the propagation of these elementary particles with waves in the math, is because that is what best fits the data from the experiments.
Now you are confusing the wavefunction with the waves that represent in the math the way elementary (and sometimes composite too like an atom) particles that propagate.
The wavefunction is the probability destribution of the elementary particles' position (and other characteristics) in all of space.
The way particles (elementary and composite) propagate through spacetime is best represented in the math of the model with a wave. This wave could model the propagation of an elementary particle, like in your case a quark, or a composite particle, like an atom.
Now you are asking whether the elementary particles that make up the atom, like the quarks, the gluons, the electrons, are represented by their own field. Yes, each of these elementary particles have their own field in QFT. Now you could interpret these elementary particles as an excitation of their own field. And no, the composite particle, like the atom does not have its own field, there is no atom field.
Now you are asking whether these elementary particles propagate as waves as an excitation of their own fields, inside the atom. Yes, these elementary particles inside the atom are zigzagging, propagating inside the atom close to light speed.
And the atom itself is propagating through spacetime, and yes, you can in the math model that propagation as a wave.
Now you are talking about the double slit experiment, where the particle as a wave is propagating through both slits, and interfere with itself. It is a common misconception that the particle itself propagates through both slits. What really happens in QM, is that the photon, for example, as a wave, is propagating, and part of this wave is propagating through one slit, and another part of this wave is propagating through the other slit. Now then, the partial waves interfere, and that destructive interference, is where you see the black on the screen. And the constructive interference is where you see the white on the screen.
Now in case of the elementary particles inside the atom, those are traveling in empty space, of course, there is interference, so you could interpret these particles' propagation like that.
Now basically you are asking whether the interference works with composite particles, like the atom, so do the atoms propagate as waves, and interfere with the partial waves, and create an interference pattern? Yes, the answer is they do create an interference pattern. Please see here:
Question about the double slit experiment 2