As Luboš Motl and others have argued, we are almost certainly not Boltzmann brains in the obvious sense: a thinking object produced by a random fluctuation within the universe as we know it. But perhaps you ask whether the universe as "we" know it—that is, as a given observer knows it—might "be" the imagination of a randomly created, thinking object in some outer world—that is, whether said observer might reasonably consider the world to be such an object's dream. This, I argue, can not be disproved.
As observers, we collect observations and form hypotheses. Collected observations are essentially just data. Hypotheses are essentially observations of our own thoughts. So as scientists, we are essentially just observation databases, or if you prefer, just one database.
All that which we know about the universe, including the fact of its existence, consists of our observations. Much confusion stems from the assumption that the universe first exists, and that we observers arise within it. But such an assumption is unwarranted. We must acknowledge the primacy of observations, given that these are necessary to establish the very existence of the universe. When we speak of our universe, we mean a certain mental model with which each of us seeks to explain and predict his or her observations.
In this context, let us consider Sean Carroll's statement: In the set of all such fluctuations, some brains would be embedded in universes like ours, but an enormously larger number would be all by themselves.
On first reading this, I imagine a real, human brain floating in a huge vacuum of space, thinking along until it freezes or otherwise dies, eventually to vanish by proton decay or quantum tunneling. But then I realise how I arrived at that image of a Boltzmann brain and how much further I have to go. Yes, an enormously larger number of lone, human-like brains would arise, but so would an even larger number of exotic, thinking objects unlike anything on earth.
When Dyson, Kleban, and Susskind touched on the problem, they didn't even go as far as the lone human brain. They gave the example of a universe like ours but with a 10 Kelvin background radiation instead of 2.7 K, and they pointed out the trouble that such a change would cause for cosmology (as if cosmology didn't have enough trouble already).
So yes, a real universe as modeled by scientists runs into trouble, and so do a slightly warmer universe and a freezing brain in space. But what would a Boltzmann brain really be like? Well, it would have to satisfy some criterion of capacity for thought, but that's it: anything more would be unlikely. Probability would far outshine both evolution and human technology in the design of such a minimal brain/computer. It would not react to the vacuum like an earthly organism suddenly cast into space. It would contain essentially no feature extraneous to the thought requirement, such as redundancy or component isolation.
Boltzmann's brain would have neither sensory input nor motor output, and anyway, there would be nothing around for it to perceive or to move. A human lacking those things would, I think, sleep very deeply. It is hard to extrapolate from humans to the class of typical Boltzmann brains, but given the requirement for thinking, we can assume that nothing would disturb its thought, nor would it feel any urge to affect its environs.
What I describe strikes me as similar to a dream. Perhaps our universe—which is to say, my observations—are part of such a dream. (Before I am accused of solipsism, note that in this supposed dream, I—the human author of this post—seem to have roughly the same kind of brain and mind as other humans, so there is no contradiction in their reaching a similar hypothesis with themselves at the centre.) Like a human dream, our world seems internally consistent—at least, mostly so—but never completely fathomable.
If the dream-universe analogy holds, what can we say about the dreaming Brain? Almost nothing. Nothing constrains it to be made of atoms or subject to our physical laws, since it exists in a meta-world. Perhaps, as in a human dream, our world somewhat resembles the Dreamer's world with a few subconscious elements thrown in, or perhaps not.
Perhaps, also as in a human dream, the Dreamer makes this world up as it goes along, in which case I see no problem with your point about unconstrained microstates of the past. Occam's razor suffices to discard the hypothesis about a malicious demon.