As far as the fractional Josephson effect due to topological superconductivity goes, the answer is yes -- you need parity conservation (of the entire junction) to see the full $4\pi$ periodicity. If electrons are allowed to tunnel into the junction, for example from external leads, the system will always jump to the ground state, restoring a $2\pi$ periodicity as a function of the phase. Hope this helps; I will try to write a full answer in the coming days.

That is an odd argument to make. Anyway, we also have general relativity which is a completely classical theory whose equations are nonlinear. Its predictions are robust and have been experimentally tested to satisfaction.

Let me clarify. Nothing stops me from adding a $\phi ^n$ term to a classical scalar Lagrangian. This is exactly what one would call a self-interaction term. Of course, one cannot interpret that term as a vertex in the Feynman diagram (as done in QFT.) Nevertheless, your statement about the fields being additive won't be true for a classical $\phi ^4$ theory since the equations of motion will be non linear.

@R. Emery that is true only for free fields.

It is incorrect to say that fields don't interact with themselves classically. One can always add a potential which contains self-interaction terms, although the notion of "self-interaction" cannot be made sense of classically.

Try Hartman's lecture notes. It assumes some background in GR, QM and QFT though.

arxiv.org/abs/1810.11563. He talks about this in the second lecture in these notes.

I have another naive question about this calculation: would the equation being about growth in $\textit{Euclidean time}$ make a difference? It would give an extra factor of $\textit{i}$ making the growth rate imaginary, right?

I'm sorry for the late comment; just now got the time to look at this. This was quite helpful. I think you mean to say "$\textit{volume}$ growth rate is constant" (of course your statement can be made by assuming the C=V conjecture, but I was looking for a pure GR answer to go towards the conjecture). Also, the assumption of volume growth being constant is fair (arxiv.org/abs/1411.2854).

Not sure if this is what you're asking, but this reminds me of something I encountered in QFT. Basically, the propagator of, say, a scalar field theory is non-zero even for spacelike seperated points (the fuzziness you're talking about reminded me of this point). But when you calculate the commutator of spacelike seperated fields, you find that the contributions from the propagators cancel out to give zero. This means that there should be no observable consequence of some perturbation on points which are spacelike separated from the point you perturbed.

You should add more information. At the very least, what kind of magnet are you using?