Regions separated by infinite potential

Is this really the case?

Yes. If you know that the particle is in a specific region $A$ at time $t$ and that the region is surrounded by an infinite potential, then this imposes a boundary condition on the time-dependent Schrodinger equation, such that the probability of finding the particle outside this region is zero at any later time $t^prime$. Formally:

$$left[forall x notin A: psi(x, t) = 0 right] wedge left[ forall x in partial A: V(x)=infty right] Rightarrow forall x notin A: psi(x, t^prime) = 0 $$

what is the difference between separations caused by a region with ψ(x)=0 and the nodes where we also have ψ(x)=0? What determines whether if the wave function passes through a region or not?

It is the infinite potential that prevents the wavefunction from passing not the nodes. If at a later time the infinite potential is removed, the wavefunction will spread outside its initial confinement region $A$. The nodes of the wavefunction at a specific time do not restrict the wavefunction at a later time. Indeed the nodes themselves can change with time deepening on the situation.

Also a final question: If we make a system with the same configuration but make the infinite potentials to become finite but very large (compared to V^ in D′ and K^, for example), can we see a similar effect or will it break down immediately? Basically, can we (more or less) separate two spatially disconnected regions using infinite or finite potentials, and if so, to what extend?

Using a finite potential, there is a finite probability that the particle will cross that barrier (This effect is known as Quantum Tunneling). The probability is an exponential decaying function that depends on the height and thickness of the barrier. As the potential becomes larger, the probability gets smaller till it becomes zero in the infinite potential limit.