I'm aware that accretion disks around black holes are formed from the swirling mass of matter that is slowly being stripped of its atoms, but what happens to it when two black holes merge? I was thinking maybe it gets eject or sucked in completely. Not sure why though. In specific, what happens to the nearby objects; objects that are both stationary and objects in motion ($v = nc,$ where $0.6<n<1$).
Accretion disks around black holes in binaries are formed when they accrete material from the other star in the system - either via a wind or if the other star fills it's Roche lobe.
If the other star is also a black hole then it cannot lose matter and so an accretion disk cannot form.
Whilst it is probable that one of the black holes could have had a vestigial disk as a result of accreting material from the progenitor of the other black hole, this is unlikely to be there when the black holes merge. There must be a reasonable lag between the formation of the second black hole and the merger. This will take of order a few 100 Myr to many Gyr according to theoretical models (e.g. Marchant et al. 2016, https://arxiv.org/abs/1601.03718) and a vestigial disk should have been accreted long before this.
Having said that, there was some interest in the idea of long-lived vestigial disks in this very context. The first BH-BH merger detected by LIGO had a very tentative gamma ray counterpart. Some did attempt to explain this as due to the presence of a long-lived vestigial disk around one of the black holes (Perna et al. 2016, https://arxiv.org/abs/1602.05140). In their model, the disk is dormant, because it cools enough to become neutral and switch off magneto-rotational-instability viscosity, but is revived by the approaching merger, resulting in tidal heating, the production of a gamma ray burst and the rapid accretion of the material into the merged black hole over the course of a few seconds.
Kimura et al. (2017, https://arxiv.org/abs/1607.01964 ) also consider this scenario, but they find that the dormant disk reactivates thousands of years before the merger, resulting in a much less cataclysmic event. The net result is the same though, accretion of the disk material and the launch of a less powerful jet and gamma ray burst
None of the subsequent LIGO BH mergers have shown any sign of a gamma ray burst...
If we just assume there are accretion disks there for some reason (pace Rob), a first order approximation is that they consist of freely orbiting matter of negligible mass compared to the black holes. This means that they will behave like the old galaxy collision simulations (commonly used as screensavers on Unix systems in the 1990s): rotating disks of points (in stable orbits around one mass) with no self-gravity but affected by the gravity of the two centre masses as they pass each other or merge. In this case you typically see streamers of points, and sometimes big splashes as large clouds scatter in all directions. In short, this predicts that the result will be a dramatic mess. The end result is typically like an elliptic galaxy, a set of shells of orbiting points with roughly spherical symmetry.
There are two main oversimplifications. The first one is likely marginal: the models don't include general relativity. But this is only important within a few Schwarzschild-radii from the holes. The second one is much more serious: accretion disks likely have much more self-interaction. They are after all based on "friction" interactions turning potential energy into heat via shear and turbulence, the emitted radiation can be stron enough to push away plasma, and the plasma tends to have complex magnetohydrodynamics. This is likely to have big effects, but I do not know how that can be predicted without just running extensive simulations. These interactions will also make the initial spherical or ellipsoidal cloud start coalescing into a disk again.
Ivanov, P. B., Igumenshchev, I. V., & Novikov, I. D. (1998). Hydrodynamics of black hole-accretion disk collision. The Astrophysical Journal, 507(1), 131 (pdf) solves half of nearly the right problem: a black hole passing through a disk around another hole. They find that as the hole passes through the disk it causes an intense convergence of gas that in turn produces shockwave heating and "fountains" of extremely hot gas. This is likely to happen when two holes merge too, when they cut across each other's disks. Even if they don't (e.g. a parallel approach), the disks will both contract and collide which is going to cause a lot of heating.
So a plausible guess is that the disks will splatter over the vicinity, sending some plasma along hyperbolic arcs towards infinity or fountaining off after being strongly heated while leaving other plasma on various bound orbits. There is a lot of virialization going on that likely produces a more tightly bound remaining disk once things settle down.
If there are two BH moving around, there's a big chance that there's no accretion disk from the start or that it is far away from the binary.
Once both BHs merged, the far away accretion disk would still be there, without much changes.
Or depending on the initial conditions (accretion disk around each BH, for example), the final matter in the disks could be partly absorbed by the BHs during the coallense process, and partly diffused away. Some parts may still be orbiting the final BH.
All scenarios are possible, it just depends on the initial conditions.