A very interesting question that has undeservedly waited too long for its answer!
I will try to address the different parts of the question one-by-one. However, excellent references are available on the subject. I can recommend Basic superfluids by T. Guenault as a simple intro to superfluidity in general, while a more difficult read An introduction to the theory of superfluidity by I. M. Khalatnikov goes into the details of the two-fluid model and is a classic.
Are these two fluids a true ''mixture'' or are is it just a model to break the behavior of the one real fluid into its two different aspects?
This is an ontological question. My practical opinion is that it is just a model that explains certain aspects of a particular physical system well, in this case Helium-4 at a certain temperature and pressure regime.
Could we separate them from each other as if they were truly separate fluids?
As pointed out in the comments, they are not two classical fluids, made of different particles. One of the fluids (normal fluid) is classical, the other (superfluid) is not classical, and they both are made of the same particles (helium atoms). One fluid exists in a particular stable state (thermodynamic equilibrium with a certain temperature and pressure) only because it is in equilibrium with the other fluid. In particular, it is possible to generate more normal fluid by heating the liquid, or cool the system by removing the normal part (more on this below). As an aside, demixing two classical fluids is already not a trivial procedure, as one has to decrease the entropy of the system (for a way to make electricity by the inverse process, see this article).
For example, if it were truly a mixture, we could make it do something that only a superfluid could do, such as pass through an extremely small opening. This would separate the fluids and leave us with just the superfluid component. Would this remaining ''pure''-superfluid still be described by the two-fluid model?
This is absolutely possible and has been demonstrated experimentally. It is called the inverse fountain effect. In order to understand this effect, it is essential to introduce a concept of a superleak. That is a material, through which only the superfluid part of the liquid can pass. A typical superleak is made of some solid porous material with tiny gaps (<100 Å).
Consider two vessels (A and B) containing superfluid helium. Initially, they are at the same temperature and pressure. A and B are connected by a superleak but otherwise isolated (in particular, thermally insulated from each other).
Now, increasing the pressure inside A will lead to a flow of liquid from A to B. However, only the superfluid part of the liquid can go through the superleak. This means that the superfluid fraction will decrease in A and increase in B. Higher superfluid fraction means lower temperature, so the temperature in A is increased, and in B decreased. In principle, it is possible to achieve a $T=0$ situation in B in this manner.
The contents of the B dewar would still be described by a two-fluid model. However, this two-fluid description at $T=0$ is much simplier that at a finite temperature, as the normal-fluid fraction is zero in this case.
Practically, the inverse fountain effect is limited by the non-ideal heat insulation between the two dewars (as well as the outside), and also by the heat (and thus normal fluid) generated the turbulence in the superleak and adjacent regions.