Is it possible for one black hole to pull an object out of another black hole? Suppose we have a spacecraft just inside the event horizon of a black hole, struggling to escape, but slowly receding into it. Another (bigger) black hole expands until its event horizon includes the spacecraft as well. This pulls both the black hole and space-craft into itself. As the spacecraft is closer to the second black hole, it receives more force, perhaps just enough to pull it out.
Is it possible that the spacecraft, even momentarily, is pulled out of the event horizon of the original black hole (although it is now trapped inside the event horizon of the new one)?
 A: The question really boils down to the dynamics of event horizons when black holes merge.  It turns out that there are some great simulations that explore these dynamics.  If one scrolls down to the bottom of this black-holes.org page one can see a video of the merging of two different sized black holes.  One can review the underlying paper and see that the actual development of the simulation was very extensive. 
The actual event horizons do move and oscillate, so the question is whether the spaceship itself has become some sort of physical element of the black hole after it has crossed the event horizon.  Since it is argued in most cases that space craft can cross the event horizon in large black holes without witnessing any sort of significant effect, although the spacecraft's mass must be considered part of the black hole's mass after crossing the event horizon, it still has some freedom of movement.
We can see from the simulation that the geodesics that define the event horizon fluctuate when the holes merge.  So if the geodesics fluctuate is it possible that the spacecraft would find itself on a geodesic that suddenly allows for an escape?
The answer should be no.  The geodesics defining the horizon require trajectories with velocities greater than the speed of light.  The spacecraft can not exceed the speed of light.  So while the geodesic its on might distort during the merger of the black hole, it is the underlying space itself that is distorting, which is not going to impart some ability to defy local laws of physics to the space craft.  As such, it will stay inside the blackhole horizon since its geodesic, while distorted, will still remain inside the horizon. 
A: In the scenario you're describing, the event horizons of the black holes will merge. Once they have merged, they can't split apart again. This is a theorem proved by Hawking and Ellis as proposition 9.2.5 in The large scale structure of space-time, p. 316.
A: An event horizon is the theoretical boundary around a black hole beyond which an observer cannot be in contact (i.e) the region from which light or any other radiation cannot (classically) escape from the inside.
In my opinion, it's NO because once an object has reached beyond the event horizon, we don't even know what happens to it inside. It can't even try to escape. Only spacetime struggles inside the blackhole because, all the possible trajectories of the object point towards the singularity. An observer would see the spacecraft getting red-shifted so much and finally disappear at the moment it nears the horizon.
Even though the spacecraft is pulled into the other black-hole during the merger or hits the singularity, we can't observe it. Instead, we can satisfy ourselves that two black-holes which are close enough could merge to form supermassive blackhole and the spacecraft is shared by both.
A: If I understand correctly what I've read from Hawking, Thorne, et al:
Once past the event horizon, space itself recedes at C.  This is the reason the redshift increases to infinity at the event horizon-- treating light as a wave, the wavelengths increase so dramatically that they essentially flatline.
When the second black hole's event horizon meets the first, and they begin to merge, anything that's left our universe through the event horizon is still stuck on the other side.  The mass/energy's probability wave includes zero possibility of it ever returning to the side on which it started.  The information about what it was is embedded in the event horizon, but I'm honest enough to admit that I don't understand the holographic principle & how this works.  The math is simply beyond my level of education.
A: The form of a black hole horizon behaves as a viscous liquid. It flattens at poles when BH is rotating, there can be waves on the surface of a black hole when it eats something and the energy of waves then radiated as gravitational waves so the surface becomes smooth again.
As something falls on a BH, its diameter increases, but not simultaneously everywhere (the BH can be so large that even light takes some time to reach the other side, so are the gravitational waves). When two BHs are approaching each other they also distend towards each other as two drops of attracting liquids, and then they fuse. The energy of the surface waves is again radiated as gravitational rays.
Closer to your question, being between two black holes would not help to escape, just the opposite, the surfaces of the both BHs expand in the direction of each other so you would be in even greater trouble. To escape from between two BHs to the infinity you'd have to fight the gravitation of both.
A: Consider a small particle such as an adventurous neutron travelling on a course directly between two equally sized black holes. The event horizons would be compressed between the black holes because they would be pulling our lucky neutron in opposite directions. So he might get right past both black holes without getting sucked in. However, based on the definition of an event horizon, once the particle falls within it, he is already on course to be trapped within the black hole forever. 
