How can we be sure that degenerate stellar matter does not bring up emergent phenomena not considered during their theoretical analysis? With the astrophysical predictions on stellar evolution (which I admit I am not familiar with in detail) I always ask myself how we can be sure that they are correct although they are extremely simplistic (they have to be, because exactly solving quantum field theoretic problems of the size of a star is outright impossible, even numerically).
Well, for the usual stars, I guess one can at least find retreat in the view that they are all gas or probably kind of a liquid, so they can be justifiably treated thermodynamically, while neglecting as irrelevant some fluid dynamics, boussinesq convection and stuff. But complex stable structure formation within these kinds of matter seems pretty unlikely.
But as soon as it gets to degenerate matter in white dwarfs and neutron stars, as far as I understand it, this gets more and more into the direction of solid state physics. And, if matter forms a lattice of one kind or another, it seems to me more likely that complex structures could possibly emerge, that cannot be foreseen in a (more or less) simple thermodynamic calculation.
As an analogy, imagine a hypothetical astrophysicist from the Alpha Centauri system who wants to understand what is going on here on earth. He knows the element composition, the temperature and pressure, and then he starts performing his thermodynamics math. He will probably find out some chemical compounds, but would he ever reach beyond the conclusion that earth is just a big primordial soup? However, we do not even need to take life into account, there is enough interesting chemistry and structure formation going on on planets and moons. Take for example Enceladus or Io, which have surprised astronomers with their structural richness. Even good old plate tectonics is a pretty complex topic.
So, how can we be sure that complex structure formation is not taking place or is irrelevant in neutron stars and white dwarfs, and that it does not influence the prediction of the stellar evolution of these objects?
 A: It is the object of physics to answer questions like yours. In your example, the way this is done is to collect up all the relevant laws of physics, build them into a computer simulation, run the simulation, and then compare the results with astrophysical behavior we can observe with telescopes and other instruments.
If the simulation does not comport with reality, it means either 1) there's relevant physics (which could be unknown!) we left out of the simulation, or 2) there's something we do not properly understand about the (known) physics we did put into the model.
For example, as pointed out by knzhou, numerical simulation of things like stellar cores and neutron stars is based on something called the equation of state (EOS) of the stellar matter: how its temperature and pressure vary as the density of the matter varies. Writing down a satisfactory guess at the EOS is an exceptionally difficult task, because at the densities of stellar cores, almost none of the simplifying assumptions that let us derive the ideal gas law (basically, the EOS for air and other gases) can be made, and second-order effects that we don't have to worry about in the case of the gas law are non-negligible.
A: 
So, how can we be sure that complex structure formation is not taking place or is irrelevant in neutron stars and white dwarfs, and that it does not influence the prediction of the stellar evolution of these objects?

These things probably do happen in both white dwarfs and neutron stars and are the topics of contemporary research.
Your question is based on a false premise. Nobody working in the field thinks that all of stellar evolution is well understood. Quite the contrary.
In the case of white dwarfs, as they cool they crystallise. This obviously does have thermodynamic implications - raising their effective heat capacties and prolonging their cooling. There is ample evidence for this in the "luminosity functions" of white dwarfs and also some more direct evidence from asteroseismological observations of pulsations. However the details are not completely understood because of the complexity of modelling mixtures of carbon and oxygen nuclei surrounded by degenerate electrons at high pressures.
With neutron stars, collective interactions are all important to the equation of state. Indeed, neutron stars are not supported by ideal neutron degeneracy pressure and an understanding of many-body, highly asymmetric nucleon interactions is crucial, but still rudimentary. Phenomena like superfluidity and pairing of neutrons is probably of little importance to the overall equation of state but probably plays a vital role during early neutron star cooling. Collective nucleon interactions also probably lead to exotic "pasta" phases at densities a bit less than that of nuclei, but evidence for this is still elusive.
And so on ...
