> What are the range and domain of the function?
The range is a state vector specific to the system you're modelling. The domain is usually space if you're trying to model a system statically, or space and time if you're trying to model it dynamically. The terminology is certainly sloppy - people say "wavefunction" to mean both "the state as a function of time/space" and "one particular state value".
> Now, what is the evidence that it actually IS the probability density of the particle? That is, so far I can't find any argument, justification, reason to conclude that the probability density of the particle is not something else. E.g., the probability density of the particle might be Gaussian.
Sure. A priori there's no reason to assume that any of this corresponds to anything physically real. The point is that if you construct these particular mathematical functions, they turn out to correspond to our observations, and furthermore every system we've been able to measure (except perhaps where gravity is involved) has a description in these terms.
> Well, the wave functions have to be differentiable and, thus, continuous, and it's easy enough to have a sequence of such functions converge to a discontinuous and, thus, not differentiable function and not a wave function.
Can you actually construct this though? Not every differentiable function is a valid wavefunction.
> Now, is h actually also a wave function? Its absolute value might not integrate to 1. Okay, scale it so that we do get 1. Now what particle has wave function h? That is, what physics does this linear combination describe? What does h in
> h = af + bg
> have to do with physics? E.g., from the Pauli exclusion principle, maybe h can't correspond to anything in physics. What is going on with superposition?
No, that can't be true - any such h is necessarily a valid state of the system. The Pauli exclusion principle comes out of a case where the amplitude is 0. This is a central part of the theory - if you found a system where h was physically impossible, that would disprove quantum mechanics.
> Can't Know. We are told that as an electron goes by, we can't know anything about it except its wave function unless we take a measurement at which time we change the wave function, what the electron might do, or some such.
I think you've got something you were told backwards. We can't generally know the full wavefunction; when we want to learn about the electron we interact with it and that changes the wavefunction (that is, changes the current state vector - of course the actual function that governs our interaction with the electron remains fixed).
> The wave function splits into two parts, say, f and g. For part f, we have a detector. For the last electron we shot, we got a detection from part f. Now, what happens to part g?
> No fair saying that part g does not exist since the Mach-Zehnder interferometer can bring such f and g back together and generate interference fringes that part f or g can't generate alone. So, part g does exist. So, what happens to it?
This is the collapse/decoherence phenomenon that people argue about. I have my views, but I can't say there's a full consensus.
> Well, three space is a big place, and we usually believe that nothing can travel, propagate faster than the speed of light. So, all the wave functions so far are still traveling, still expanding. Right? What happens to them?
If you fire an electron off into the distance then yes, it keeps travelling. Just like if you fire a cannonball whose position is described by f(x, t) = [1 if x == 2t, 0 otherwise] then yeah, that cannonball keeps going forever. This is normal physics?