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Mike Wrecka in regards to point A, many people believe that something to that nature will eventually happen (The Big Freeze, as it's been dubbed), though our knowledge of the history of the universe tends to disagree thus far. Meaning, the very fact that planets and galaxies exist at all is because of gravitational discrepancies from the Big Bang. If everything was distributed evenly, then heavier elements would have never crashed into/combined with lighter elements and formed any sort of order. The concept of planetary evolution itself flies in the face of thermodynamics, too, but that's neither here nor there. But to answer your question, matter is not equally dispersed, and likely never will be.
As far as point B is concerned, there are differing views. I think it was you who mentioned the Doppler Effect, that the universe is expanding (we know this by viewing the red shift of other objects in space, which indicates that things are moving away from us). The traditional scientists believes this is the ripple effect of the Big Bang, that the force of it caused everything to continue in motion, with no force to slow it down or stop it. Others believe there are white holes, which are pumping matter into the known universe and causing it to expand. Truth is, we don't know. Or at least, I don't know. In terms of space being barren, all space is barren. That's what I mean about the fundamentally flawed understanding of what space is - space is nothing, it's an absence. So yes, assuming the Big Bang happened, matter is moving into an infinite absence of matter, unless there are eventually other universes (which M Theory suggests). Again, we are still figuring out things like dark matter and antimatter, so our perspective of these things is extremely limited at this point.