I don't know the most general form of Sturm Oscillation Theorem, but it may be worth looking at "Oscillation Theory" by Kreith which contains results of this nature. Additionally, oscillation theorems are proved in "Maximum Principles in Differential Equations" by Protter and Weinberger. Both of these books should contain further references which can also be investigated.

You are correct that the link does not necessarily consider solutions in the T\"ackland class, but it may be worthwhile to consider how the argument could (potentially) be adapted to allow the solutions to be in the T\"ackland class (since the argument does not simplt rely on integratinf the convolution of a fundamental solutions, and some initial data).

Check $u_2$ at $t=0$, the time where nonuniqueness for $u_2$ is an issue.

Reasonable idea. I'll have a think about it (sorry for the delay in response).

So, regarding existence of solutions to the heat equation, with any smooth enough initial data (without growth restrictions), the link regarding existence (which guarantees that there exists a solution to the heat equation with initial data \psi (x) = e^{x^2\log{(1+x^2)}}$ adresses your query, with regard to the standard fundamental solution to the heat equation. Perhaps it is worth considering if there are alternative "non-standard" fundamental solutions to the heat equation, and what properties these might satisfy (or if a uniqueness result holds).

There are numerous books covering local existence theory for boundary value problems for second order quasi-linear (and hence, semi-linear) parabolic partial differential equations, as described above. The classics by Friedman "Partial differential equations of parabolic type" and Ladyzenskaya, Ural'ceva, Solonnikov "Linear and Quasilinear equations of parabolic type" contain relavant theory. A More recent book by Lieberman "Second Order parabolic differential equations" is also pretty good, amongst others.

To apply Laplace's method, I am under the impression that one requires certain regularity/non degeneracy conditions on $a(x)$ near to $x=0$, as well as global integrability conditions on $a$ ... is this implicitly assumed here or not? If so, could it be stated in the question, and if not, could you refer me to a reference where Laplace methods are considered with no regularity/degeneracy requirements are placed on a(x) near to 0? Cheers

Could you be more specific with regard to the weird behaviour?

Here, the singular coef. in your diff. eq. doesn't concern whether or not a comparison principle can be obtained (these depend on existence of parabolic weak maximum principles). This is due to the sign of the zeroth order term in the diff. eq. (the coef. of the first order term in the diff. eq. does not effect a max. principle argument). Provided that you specify appropriate boundary conditions on the solutions, notably for $(x,t)\in ([0,1]\times\{ 0\})\cup (\{ 0\}\times (0,\infty ))\cup (\{ 1\}\times (0,\infty ))$, as well as regularity on the solutions, then a comparison principle exists.

what are the regularity conditions on $p$, $\Theta_0$, $u$ and $v$?

It seems that this should be for some $\epsilon >0$, not every.

Could you please specify the domain on which you want to solve the final quasilinear pde, the regularity required on the solution and if the domain is unbounded, the growth conditions on the solution. Cheers

It may be worth having a look in the book "Second order parabolic differential equations" by G. Lieberman, however, it may not. Good luck.