I don't fully understand what phenomena you refer to. However, as an aside, free will probably is an illusion. It seems like you have contemplated this, so I don't need to elaborate.
It is only illogical from a certain point of view. The ideas of quantum mechanics have explained, to a certain extent, the curious findings of the double-slit experiment. One of the limitations of logic is that it tends to be deductive always, that is we conclude things from certain premises. When, in reality, all of human discovery is essentially inductive. That is, we formulate laws based on observations. This does not mean we should abandon reason, but that we should remain flexible in our concepts of what appears logical or illogical.
What is psi? If it is some kind of new-age spiritual theory, than scientists should not be talking to people that are researching psi. It is only a waste of their time.
I do not think so. Psychology is defined and determined by the neurological processes. Thus, psychology is really a labeling system for different patterns of neuron firings and brain hormone levels, etc. For example, the psychological phenomenon called 'anger' has a measurable, albeit not fully understood, chemical signature. Scientists find that neurology is more fundamental and enlightening than conventional psychology.
However, all of this is academic. Why would the next revolution in science come from neurology or psychology? Certainly great innovations in those fields are being made, but discoveries in quantum mechanics are more fundamental than discoveries in psychology or neurology are, since both of those are dependent upon the workings of quantum mechanics. For me, the importance of a scientific discovery can be determined by how fundamentally it changes our understanding of the world. Therefore, I think that modern physics holds the greatest potential for revelations.