How can a singularity move? When an object approaches the event horizon of a black hole from the point of view of a remote observer time of this object freezes. (Does it also stop? Since it has a finite speed even at the event horizon?)
But the whole black hole can still move through space even though it's inside event horizon. Is it because it is not affected by the gravitational potential it generates? Or maybe is it because once it becomes a black hole it becomes more of a spacetime disturbance than a material object and thus can travel like gravity waves do (but slower) dragging the singularity (that is the source of the gravity) within it? Or is there some other reason?
 A: An object falling into a black hole freezes from the point of view of a remote observer. This only means that it stops moving with respect to the black hole. In other words, the black hole and the object will start moving together. There is nothing that prevents the singularity from moving, and there is nothing that prevents the object from moving. The only requirement is that they move together.
Additionally, having a moving black hole is very simple. You just need to move with respect to the black hole. Motion is relative ;)
Edit:
You ask how is it possible for an object frozen in time to move in space. Nothing prevents something with 'frozen time' to move in space. Indeed, if you think about it, everything that goes at the speed of light has its time frozen. Going faster in space means that you go slower in time, so the object near a fast moving black hole would experience an even slower time. You would see such object crawling even slower towards the event horizon, compared to the case of immobile black hole.
And it would make no difference if the object is in the front or in the back of the black hole, apart from the fact that, since the object is getting closer to the event horizon, if it is on the back, it will move slightly faster than the black hole, and if it is on the front, it will move slightly slower. But the rate at which it gets closer to the event horizon would be the same.
A: When an object is near the event horizon then it will stop according to an observer who is far away.

Does it also stop? Since it has a finite speed even at the event horizon?

No, in the reference frame of the object it will take a finite time to hit the singularity according to classical general relativity. For more details you can refer to page 254 of General Relativity An Introduction for Physicists
By Hobson, Efstathiou and Lasenby or any other general relativity textbook.

But the whole black hole can still move through space even though it's inside event horizon. Is it because it is not affected by the gravitational potential it generates?

I do not understand why you think that a black hole can't move. In fact, an observer who is falling towards the black hole will see the event horizon and black hole moving.
