Could an object, assuming possible, escape black hole by travelling above constant speed? I am always hearing that after travelling inside a given BH's event horizon, the escape velocity is greater than c, which makes it impossible to escape. 

Suppose an ship is inside a BH. Assume it is possible to travel faster than c. If the ship applied the faster-than-light travel, would it be able to escape a black hole?

Furthermore, it seems to me that above event horizon, the escape velocity is less than c, while inside event horizon, the escape velocity is greater than c. This leads to a further question: Does the escape velocity increase the closer you get to a BH's singularity?
Assume it is possible to travel faster than c


We normally cannot use physics to answer questions about scenarios that break the laws of physics. But for this scenario, we can say that if the ship can travel backwards through time, then it can escape from the event horizon. Just about every FTL scheme can be used for backwards time travel, and as far as we know, backwards time travel is physically impossible.

Rather than thinking about this problem in terms of escape velocity, its better to think in terms of worldlines. The velocity of a body depends on the observer measuring it, and which coordinates they're using, and dealing with that stuff near a black hole gets tricky. The usual Schwarzschild coordinates break down at the EH, and different observers measure quite different things, depending on their location and motion.

Inside the event horizon of a Schwarzschild black hole, all worldlines travel towards the singularity, without exception. As you move towards the future, you move towards the singularity. So the reason you can't escape from inside the EH is that there are no paths that go outside the EH. Unless you can magically travel backwards in time, so you can follow an in-falling worldline in reverse.