No, it’s about the gravitational field within a shell-like distribution of matter, which is exactly zero; it wouldn’t ‘pull’ on spacetime anymore than it pulls on matter inside.
Besides, the idea of gravity ‘pulling’ on space in this way is misleading in two ways: first, gravity couples to—jargon for: is caused by and influences in turn—mass, or more accurately, stress-energy, a somewhat abstract quantity that contains the local mass/energy-density, momentum flows, pressure and shear stresses. In empty space, this quantity is zero, leaving nothing for gravity to coupld to.
But things aren’t quite that simple, because in an important sense, the gravitational field is spacetime. Basically, the gravitational field tells us about the proper notion of distances between events—that is, their spatial as well as temporal separation. It’s essentially the set of rulers and clocks you cover space with in order to determine how far, in both space and time, it is from one event to another. The fact that this set of rulers and clocks is not everywhere uniform is what causes gravitational effects—the geometry of spacetime changes, and with it, the notion of a straight line, so in the vicinity of mass (or rather, stress-energy), motions will deviate from straight in a way described by the gravitational force law.
But then it turns out that even in empty space, the gravitational field need not be intrinsically zero: there may still be an overall positive or negative curvature. This intrinsic curvature may be either negative or positive, and can be captured in a quantity called the ‘cosmological constant’, as mentioned by Asympotically Fat. Our universe, seeing as how it is currently expanding in an accelerated fashion, will ultimately approach just such a state: an empty universe (all the matter having been infinitely diluted) with positive curvature, known as the de Sitter-universe.
Anyway, all this just to say that gravity shouldn’t be thought of as a force ‘pulling on’ spacetime, as if you took a sheet of rubber in both hands and gave it a good tug; rather, it’s a property of spacetime itself, which is ultimately determined by the configuration of matter (read: stress-energy) within it. Of course, even this picture is further complicated: the gravitational field itself has nonvanishing stress-energy, so it is a source of gravity—this ‘self-coupling’ makes the theory nonlinear, since you can’t just add the gravitational fields of two matter distributions together, but have to take into account the gravitational fields sourced by those gravitational fields, and so on. But this just as a final parentheses.
In general, they’re not. The shape of the gravitational field depends on the matter distribution—for instance, there are also gravitational fields corresponding to plane wave solutions.
This concerns not the accelerated expansion our universe is currently undergoing, but the expansion it underwent in the first few instants after the big bang, generally called ‘inflation’. This inflation is predicted to have left a characteristic signature in the cosmic microwave background due to the gravitational waves it produced; the BICEP-2 team has claimed to have detected the corresponding fluctuations, but now some have suggested that the claim was erroneous due to an incorrect modelling of interstellar dust effects.
However, inflation touches on another of the OP’s questions which I didn’t get to earlier, namely whether it’s possible to have different ‘big bangs’ in the same spacetime. It’s here a bit of a question of how you define ‘big bang’: if you take it to be the point of origin of our universe, as in ‘classical’ hot big bang theory, then no, this isn’t possible—our spacetime is like the surface of a balloon, all of which originates from the big bang.
But inflationary theory has led to a paradigm known as ‘eternal inflation’. Basically, inflation is caused by a hypothetical field, the inflaton, which, well, inflates; but there is always a slight probability for it to undergo spontaneous decay, in which case we get a region in which inflation stops. This region then undergoes reheating, which means, basically, particle creation—so in this post-inflationary region, sometimes called ‘bubble’, we have what looks like a big bang—a hot fireball creating a shower of particles that, as the whole thing cools, coalesce into complex structures, possibly including some like those we see in the universe.
In this scenario, due to the fact that there is always some probability for bubbles to nucleate, there will typically be many big bangs—and since the whole thing goes on forever, infinitely many of those. The space between most of these bubbles will typically be expanding exponentially, so it’s impossible for one to influence the other. But it’s in principle possible for a bubble to nucleate such that it’s within the lightcone of another bubble, in which case both universes will collide—what happens then is somewhat difficult, and depends on things like the vacuum energy of each bubble (a lower vacuum energy in one bubble will cause the other bubble to decay to this lower level, as well, for instance). Such bubble collisions can, in theory, leave observable evidence, again in the form of fluctuations in the cosmological microwave background, but none has as yet been observed.