Does a black hole move through space? What happens to other things around it? I think a black hole would move through space like stars and galaxies. But then the problem is, what will happen to the event horizon and photon sphere orbiting black hole? If a black hole moves through space does not it allow photons orbiting around it to escape or fall into it, so the black hole should always lose photon sphere?
Same for event horizon, does it move along with blackhole? How can it move? Doesn't it allow some photon at the event horizon to escape like a comet trail?
 A: Black holes moves just like all other matter.  The photon sphere, the event horizon, just like the inner-most stable circular orbit, etc, etc. all move along with the BH---as they are defined relative to the BH rest-frame, i.e. their relative velocity is zero.
Having the black hole be moving doesn't really effect the of (e.g.) particle trajectories much: it's very rare that a photon becomes trapped (even for a short time) at the photon-sphere, and it would be similar rare for a moving black hole, but the trajectories of the photons would simply be slightly different (in particular they would need have a velocity component in the direction of the black hole's motion.  But there's definitely nothing strange or exotic happening here, and based on the fundamental principle of Lorentz invariance (things being the same from different velocity reference frames) nothing exotic is allowed to happen.
A: Assuming the BH had a stable photon sphere.
I think you may have a point. Specially, for the photons that are orbiting the black hole and are moving at a time in opposite direction to that of the black hole.
There must be a light speed lag between actual movement of the singularity, and evolution of the curvature at the edges of the event horizon. So the photons on the rear end of the event horizon are travelling at light speed c (note c is constant), but the EH has moved a little forward, and so they should be outside EH for a moment which should allow them to escape the EH. 
Suppose there is a 1 nano second lag between actual movement of the singularity, and evolution of the curvature at the event horizon. The photons are moving as per the EH of singularity 1 nano second ago. Now all of a sudden, (1 nano second later), the EH has moved by (say) 1 micrometer. The photons as per older EH should fall outside EH and thereby escape the EH.
Same can apply to photons on front end of EH and they should fall into the black hole.
I do not know if and how the mathematics accounts for this.
I can understand a north/south satellite around earth because the velocity of the satellite can remain same relative to earth in both directions because, its speed does not have to be constant. But light going north/south around a black hole, its speed has to be same relative to the singularity, at the same time be constant in both the directions, which is hard to grasp.
A: Well we don't know yet, 
 However if you consider Hawking Radiation which shows that quantum effects allow black hole to emit black body radiation it can be theorized that Black holes and their event horizons can move.
And if it is correct that black holes are collapsed neutron stars, and being as we know stars move, this is further evidence that black holes do move.
Additionally if indeed galaxies move, and black holes are found in the centers of galaxies even more evidence for consideration.
I don't think you can assume that a Black hole will emit photons just because it moves, it's gravity field will move with it, still holding the photons.
As black holes are most likely moving and we do not observe photon radiation, this would seem to indicate that there is no "photon comet tail" resulting from black hole movement, or we would have seen it by now.
Suppose two black holes are moving towards each other on a collision course and one is larger than the other, you might think that the larger one could pull photons away from the smaller one. Not likely, the collision will happen and still no photons escape, you will just have a larger moving black hole. A feather and a stone fall at the same speed in a vacuum.
Now if a giant black hole is a spining moving collapsed neutron star then, that "comet tail" may be observed as the spiral arms of a galaxy. 
A: It is theorized that black holes are only at the center of galaxies.
These galaxies are moving relative to each other. By the expansion of space between them, this means that the object at the center relative to the space around it is not moving at all. Hopefully this helps with your question.
