Microcanonical ensemble, ergodicity and symmetry breaking
First, I strongly advise you to read sections 24 and 25 of Tolman's excellent book on statistical mechanics. My answer will mainly go along the lines of what is in the book.
The ergodic hypothesis states that a system will eventually in some time interval $T$ visit all the states compatible with a given energy constraint. As you said, this implies an equivalence between the microcanonical ensemble average and the time average.
This hypothesis was introduced by Bolztmann and Maxwell in an attempt to give a physical (non-statistical) justification to statistical mechanics. The reasoning is that statistical mechanics would give a way to calculate averages over time of quantities which in turns links with averages over many repetitions of an experiment. Then, if the ergodic hypothesis could be justified with the laws of classical mechanics, statistical mechanics would not need to introduce any additional postulate (the postulate of equal probability or maximum entropy).
We now know that the ergodic hypothesis is flawed for two reasons :
- Classical mechanics shows that systems do not explore the entire phase space corresponding to a given energy constraint, but only a subset of it. The trajectories can be very large but do not pass by each and every point. There is a quantum version of this statement.
However, if we let small perturbations from the environment affect a system, something like the ergodicity can becomes be true. This assumption is realistic because no system can ever be totally isolated. To go back to your example, a paramagnetic (or non-ferromagnetic) material would look like it is ergodic. It would explore most of the available states because of the small electromagnetic perturbations that affect it. On the other hand, a ferromagnetic material would never look like it is ergodic because small perturbations cannot make the magnet change its orientation. So you are right: systems in which there is a large energy barrier between states are definitely not ergodic.
- Even in cases where something like the ergodicity holds, the time of recurrence $T$ can be very large, in fact be larger than the age of the universe.
Finally, some of your questions are more oriented on the concept of spontaneously broken symmetry. You may want to look at some other answers on this specific problem, for example this one.
EDIT : This article also gives good explainations, more specifically on the impossibility of strong ergodicity.
He [Boltzmann] put forth what he called the ergodic hypothesis, which postulated that the mechanical system, say for gas dynamics, starting from any point, under time evolution Pt, would eventually pass through every state on the energy surface. Maxwell and his followers in England called this concept the continuity of path (3). It is clear that under this assumption, time averages are equal to phase averages, but it is also equally clear to us today that a system could be ergodic in this sense only if phase space were one dimensional. Plancherel (14) and Rosenthal (15) published proofs of this, and earlier, Poincare (16) had expressed doubts about Boltzmann’s ergodic hypothesis.
The way I see it, your questions are closely related. Imagine a system at high temperature, rapidly exploring a lot of its phase space. You might say the ergodic hypothesis is correct in this situation. Then you start reducing the temperature, and the energy, and it may happen that for low energies there are two regions in phase space with the same energy, but far apart. Then the system, which is moving about chaotically, will be trapped, as the temperature falls below a certain limit, to one of these regions. This could be seen as a phase transition and the ergodic hypothesis would not be true in a naive sense (because not all microstates with given energy would be equally likely, only those in the connected component that was "chosen")
As everything else in physics, a microcanonical ensemble is an idealization, useful to get started and to build some intuition.
Classical physics, where ergodicity may be invoked for simple enough systems, is also an idealization. In quantum mechanics, which is the more accurate theory, the notion of ergodicity has not even a place.
Thus you are right - there are no microcanonical systems. At equilibrium, the typical systems are grand canonical.