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Due to gravity and movement an object can be accelerated (positive/negative) and slingshot into another direction.
This has been used various times, but as far as I know with orbital objects like a Jupiter.
It could be the case that black holes have far too much gravity to be used for swing-by or that one would have to get too close.
Would it be possible to use black holes for the same purpose? For example for interstellar flights.
This is actually a wonderful question for physics, you asking about rather specific aspects of astrophysics. I'll give it a try anyway.
Oversimplified, seen 'from a distance', a back hole is just another massive mass point in space. So the answer here is yes, it can be used.
The problem is, however, that you need to keep 'a distance'. There are two issues here: As you mentioned, you want to be able to escape. So you must not cross the event horizon of the back hole. Below this 'horizon', all trajectories of objects, even at light speed, will eventually fall back to the black hole. This brings you to the second issue: Relativity. A fly-by near a black hole can theoretically accelerate you to virtually the speed of light, at least temporarily. It depends, whether you actually want this, including all side effects.
Slingshots work by gravity interaction between a massive body in motion and a second body in motion. Any massive object can be used.
The problems with a black hole are (1) avoiding the capture range, (2) surviving the radiation, (3) surviving the accretion disk contents, (4) the extreme distances involved.
Keep in mind: black holes can have tremendous distances with gravitational effects. Distances which can be measured in light years in some cases, and thousands of AU even for smaller ones.
The strength of the boost is a factor of how close one gets, and the correct angle of insertion relative to the motion.
The further out, the less the acceleration/deceleration. But also, the further out, the less the radiation from the accretion disc collisions, the less dense the accretion disk, and the less Hawking Radiation.
The immense distances also require immense amounts of time.
Yes, as long as the orbits are long and far enough.
The problem is that relativistic mass IS a gravitational mass. The faster an object moves, the heavier gravity-wise it is, and while at speeds below, say, 0.5 c this mass gain is insignificant, as you're approaching speed of light about all of acceleration energy is converted into mass - and as result, into gravity force.
So, if you try to approach a black hole through elliptical orbit where the "near pass" speed would exceed speed of light if counted in Newton and Kepler way, obviously that won't work. Instead of speed that would counteract the centripetal force of gravity of the black hole, you'll start gaining mass, and that will make the black hole's gravitational pull on you so much stronger. "Falling", instead of speeding up you keep gaining weight, and so the gravity pull grows not only with waning distance but with your kinetic energy growing - until the inevitable demise.
While you can "slingshot" into another direction you cannot do a proper gravity assist in isolation; you would need a reference point from which the black hole appears moving so it's velocity is added to the probe after the flyby. Within the solar system this reference point is the Sun and Jupiter is moving, so you can use it to move faster relative to the Sun.
From a black hole centered viewpoint an unpowered flyby doesn't result in leaving the system with higher speed.
But by using a powered flyby you can gain a lot of extra velocity. Let's assume you arrive with 100km/s hyperbolic excess velocity (that's 100*100 = 10000 units of kinetic energy), near the black hole your speed at the perigee is 3000km/s, you do a 5km/s burn, so you gain 3005$^2$-3000$^2$ = 30025 units of kinetic energy making you have 40025 units, doubling your speed to about 200km/s when you leave the system.
