It looks like it steals energy from an object in motion, in this case a black hole. (Presumably because this is a dense object so you can get closer to it.)

Existing satellite slingshot maneuvers steal energy from the thing they slingshot around -- every satellite that slingshots off earth steals a very, very small amount of energy from the earth. The closer you get to the center of Earth's mass, the more energy you can steal. It's just that in Earth's case there's all this "atmosphere" and "lithosphere" garbage preventing you from snuggling up right close to the center.

With a black hole, all that stuff is compressed, so you can get closer and steal some of its kinetic energy. (If I understand the concept correctly.)

Technically this steals energy from the binary pair, not the black hole itself, but yes it's a potential power source for a sufficiently advanced civilization.

Yup, there are a variety of ways. Probably the most dramatic one is the 'black hole bomb', which you should definitely look up~

Kinda? This let's you remotely steal inertia from rotating black hole pairs or inertia from a black hole traveling towards you.