Can spacetime exist in the absence of matter and energy? I'm pretty sure Ernst Mach would have said that spacetime cannot exist without matter in it.
But I'm also pretty sure that a black hole can be described as a self-sustaining gravitational field, which is (I guess?) a case of spacetime sustaining itself without the matter meaning much anymore.
So, what is the current theoretical view on this?
 A: The first problem we run into in answering this question is definitional. What does "matter and energy" mean? The Schwarzschild spacetime has a zero stress-energy tensor everywhere, and yet we characterize it by its mass $m$. A distant observer measures the black hole's gravitational field and says, "Yep, it has mass $m$" (i.e., its Komar, ADM, and Bondi masses are all $m$). Physicists traditionally don't include electromagnetic fields as matter, but relativists call them "matter fields." Gravitational waves don't have a definable contribution to the stress-energy tensor at a given point, but they do have energy if you average them over a wavelength.
A second problem is simultaneity. Suppose for the sake of argument that all matter in our actual universe is eventually going to end up in black holes. (We're pretty sure that this won't happen, but it's not logically impossible, just not likely given what we know about astrophysics and cosmology.) Clearly right "now" our universe contains matter. At some "moment in time" a gazillion years in the future, if we imagine that all that's left is black holes, we could say there will be no matter, in the sense that the stress-energy tensor is identically zero everywhere. If we believed that spacetime could only exist in the presence of matter, then we would have to say that at some "point in time," spacetime would have stopped existing, since the last piece of matter went into a singularity and was no longer present on our spacetime manifold. But the scare quotes around "now," "moment in time," and "point in time" remind us that we can't fundamentally define these times in the sense of universal simultaneity. Relativity doesn't have simultaneity. When matter falls into a black hole, it's valid for a distant observer to say that the matter never makes it past the horizon.
As a final absurd possibility, suppose that Wheeler's geon idea turns out to be valid beyond his wildest dreams. (This is not considered likely anymore, but it's not obviously logically impossible.) We can describe electrons, quarks, etc. as excitations of the gravitational field. We could then say that right now, in the room where I'm writing this, there is no matter whatsoever. The stress-energy tensor is identically zero everywhere, and there are no matter fields, only gravitational waves.
A: Yes. To summarize the comments, there are a lot of physically interesting vacuum solutions to Einstein's field equations, including the eternal Schwarzschild black hole.
If you buy into the idea that gravity is mediated by gravitons, then this idea has a bit more meaning. You can imagine a bunch of gravitons just floating around. As the quanta of space-time itself, these gravitons could exist independently of any source, just like gravitational waves. So the particular configuration of just space-time itself could contain information.
A: There are two forms or states of "space" 4-momentum space and spacetime. The quantum vacuum is an example of both spaces mutually transforming via the uncertainty relations. In other words at the quantum level, spacetime exists as an ensemble of tiny  intervals whose longevity is the same as the virtual particles associated with them.
Thus spacetime is generated by the quantum vacuum. This is my opinion. I have worked on Fourier transformation of spacetime. 
A: The answer is "no": We know that vacuum cannot exist in the classical sense. 
If you mean "solutions to Einstein equations" rather than "spacetime", then yes (there is the trivial solution...) But this has noting to do with reality, not even with "potentially possible/imaginable reality", assuming that the laws of nature we know for sure to exist, do exist.
