How can quantum wavefunctions be smooth/continuous when particles are created/destroyed/changed?
Because particle creation etc merely involves a change of configuration. There is no magic. In gamma-gamma pair production you start with two photons, and you end up with an electron and a positron. Each has a wave nature, as evidenced by things like electron diffraction. When you annihilate an electron and a positron you get two gamma photons, and they have a wave nature too. Note that in atomic orbitals electrons "exist as standing waves". It's like you change the wave configuration from an open path to a closed path.
My (admittedly limited) understanding of the Schrodinger equation tells me that the vector differential operators are only meaningful over a differentiable phase space.
All you need to understand is that the Schrödinger equation equation is a wave equation.
For example, if the dimensions of my phase space are the Euclidean coordinates of N featureless point particles
There are no point particles. The E=hf photon has a wave nature, and so does the E=mc² electron. It's field is what it is. It isn't some speck that has a field, it is field. Think standing wave standing field, and there ain't no billiard ball in the middle. And if anybody objects to that, tell them it's quantum field theory, not quantum point-particle theory.
then it makes sense that you can construct a 3Nary field of some (algebraically) vector quantity
Yes, you can construct a standing field from a dynamical wave. That's what gamma-gamma pair production does. Then we talk about vector fields:
Public domain image by Fibonacci, see Wikipedia.
But if the wavefunction allows particles to be created, then we might now have a future state in a 3(N+1)ary field. In the finite case the interpretation breaks down.
Sorry, I don't know what you mean by this. But can I offer that the electron's electromagnetic field is dynamical, hence its magnetic moment. And its helical motion through a magnetic field. And the Einstein-de Haas effect which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics."
Now, I know the wavefunction operates on an infinite-dimensional phase space, and if I strain my imagination I can just about conceive an association between this object and the particles in the universe (if I take it for granted that all particles are everywhere and the particle field is the sum of these, okay)...
You should read up on weak measurement and stuff like this by Jeff Lundeen:
"With weak measurements, it’s possible to learn something about the wavefunction without completely destroying it. As the measurement becomes very weak, you learn very little about the wavefunction, but leave it largely unchanged. This is the technique that we’ve used in our experiment. We have developed a methodology for measuring the wavefunction directly, by repeating many weak measurements on a group of systems that have been prepared with identical wavefunctions. By repeating the measurements, the knowledge of the wavefunction accumulates to the point where high precision can be restored. So what does this mean? We hope that the scientific community can now improve upon the Copenhagen Interpretation, and redefine the wavefunction so that it is no longer just a mathematical tool, but rather something that can be directly measured in the laboratory."
but I cannot conceive of a way this wavefunction can be continuous when particles are created or destroyed.
Conceive of it changing direction. From straight to curved, or vice versa.
...and if it isn't continuous, how can we apply differential operators?