Can the UAV fly in the space station? Can birds also fly in the space station? Can the UAV fly in the space station? Can birds also fly in the space station?
Birds can bump around the space station.
 A: There is a video, how pigeons are doing in zero g. It was not on the ISS, but in a zero-g flight, so they had only about 30 seconds to adapt. Compared to this, they did it really well. Going suddenly in zero g, not knowing what is happening, we had probably much more problem with the positioning.
Most propellers can work also backwardly, but they are - as the other answer says - much lesser effective. But they could work. Fortunately, not needing to balance the gravity, a lesser effective propeller is enough.
If both the firmware of the drone and the electrical engines of its propellers allow it, it could work. However, most drones were planned never considering the possibility that once they would fly in zero g, and most of them has a completely closed firmware, making practically impossible to extend it.
Thus, probably most drones could not be made working in zero g, but it has not a theoretical, only a software cause.
A: It would take a special UAV. Earth-bound UAV's are designed not only to lift up against gravity, but to use lift-vector steering to travel horizontally.
To travel anywhere in a weightless 3-D environment, I would design it with rotors to travel not only along the Z axis, but along at least one of the X and Y axes.
Alternatively, it would have to be really nimble about rotating its one lift vector into the necessary direction. Then there's the additional problem of how to yaw in place.
A: Almost all of the drones and quadcopters do not have a mechanism to exert downwards thrust. They simply reduce their upward thrust and depend on gravity to reduce their altitude. To fly in a zero-gravity environment, you would need a heavily modified UAV that can provide trust in both directions. This would be rather difficult as traditional propellers are designed to push air in one direction.
Similarly, birds would also have to drastically modify their technique to be able to fly in zero g. It would be very difficult for them at first but definitely not impossible.
A: As far as aerodynamics is concerned, yes as long as the UAV was able to produce lift both up and down, because in space there is no gravity so there is no constant downward acceleration to battle. Therefore, 100% of your lift would be used simply for maneuvering but that also means that to move "down" it is no longer enough to just reduce your upward lift; You actually now need to produce downward lift to move down.
This also holds true for birds and would allow birds superior maneuverability compared to humans and lower energy consumption than while on Earth.
However, sensors are a different story. Inertial measurement units that use gravity as an absolute reference would need to use something else. Birds have eyes, are adaptable, and smarter than UAVs so not as much of a problem for them.
