What is the region of space that exists at the center of mass of merging black holes?
You have some region of space (lets call it the 'center') that is surrounded by black holes (BH), such that their individual event horizons (as spherical surfaces) completely enclose the center.
The 'overall' event horizon (EH) of the spacetime are not just intersecting spheres. A spherical event horizon exists for a spherically symmetric blackhole, but as soon as you place two blackholes together, the event horizon is no longer spherical. That is to say that the geometry gets more complicated. This video (which could be better rendered...) gives a good example.
Event horizons are defined (see, here or here) based on causal interaction. Mathematically, when geodesics can bi-directionally connect two regions. Obviously this 'center' region is no longer connected to the outside universe, so from an outside observer, there is just a single (complicated) EH on the outside of all of the collective BH system. An observer in the inside region could probably see an EH all around them---with everything in all directions becoming extremely redshifted. (This is analogous, but clearly distinct, from the horizon of the universe, which we also see in all directions.)
The space in the 'center' would be completely "normal" --- but the same is true with space within event horizons in general. An observing passing an event horizon to the "inside" of a BH doesn't experience anything wild (except perhaps very strong tidal forces), and they wouldn't in this central region either.
A few basic observations:
Black holes, even supermassive ones, are pretty small. The supermassive black hole at the center of the Milky Way has a radius of just 41 light seconds. The size of a stellar black hole is a fraction of a light millisecond. So, in very short order after coming into existence as an isolated region of normal space (less than a millisecond), the "center" will receive no light from the outside the "center".
While tidal forces near a supermassive black hole are modest, tidal forces near stellar sized black holes in the process of merging are huge. And, while it isn't theoretically impossible for enough supermassive black holes to converge in the manner needed to create a "center", I suspect that the probability of that happening more than once or twice anywhere in the entire history of the universe has got to be pretty much negligible.
Unless the shell black holes are absolutely perfectly balanced in mass, everything in the "center" will be pulled strongly towards the most massive part of the shell. All the mass-energy in the "center" would be sucked up before the last bits of vacuum were absorbed into the event horizon. While Birchoff's theorem theoretically applies to "shell"-"center" systems, there is no physically possible system that could really exist that would be so perfectly balanced. Moving black holes aren't amenable to that degree of precision engineering.
I can't imagine any way that the "center" can be stable. Any set of dynamics that would lead to the formation of a "shell" as the black holes converged towards a common center of gravity would moments later expand to absorb the "center", which absent a wildly improbable number of simultaneously and perfectly converging black holes, is going to have a maximal dimension on the same order of magnitude as one of the pre-convergence black holes (i.e. a fraction of a light millisecond). Given the duration of the black hole merger event measured by LIGO, it is hard to imagine a "center" existing for more than a matter of few seconds (if that).
While the "center" is "normal" space in a technical sense, it is still a highly remarkable and short lived unstable environment, that is certainly not an "ordinary" part of space even if it is technically not exotic.
It is theoretically impossible for anyone outside the "shell" to observe in any way what is going on in the "center". There is no way to empirically test the predictions of GR in this domain that I can imagine.
In the time leading up to the formation of the "shell", anyone in the soon to be "center" region is going to experience truly intense and weird gravitational lensing as black holes converge on the observer from all directions. This would tip off any observer in the "center" region who understands GR as to what is about to happen, so no sufficiently knowledgable observer could fail to know that they are in an "center" rather than a larger universe.