When a (perfect) ball is thrown at a (perfect) car, it will bounce off with the same velocity, v, but with its direction reversed. If the car is moving towards the car with some velocity V, when the ball bounces off it will have a velocity v + 2V. (This is easier to understand from the passengers of the car, who will see the car arriving at v + V and bounding at -v - V, or -v - 2V relative to the ground).

In the ball-car collision the electromagnetic forces are the ones responsible for changing the direction of the ball. But in the binary system, it is gravity. In particular as the shuffle enters orbit around the "incoming" star, the stars gravity will mostly pull it forward when it completed half orbit.

I hope I make sense.

Overall, such a joy to read this paper. With basic physics it makes you dream of sci-fi...

Your ball morphed into another car at the second sentence. Surely an effect of quantum entanglement :)

Figure 1 shows one possible mechanism. It's basically a gravitational slingshot using a binary star system. A test mass comes out with more kinetic energy than before and the binary star system's radius decreases, releasing gravity waves at the same time.

A large mass is dropped on the right trajectory between the two, the gravitational forces slingshot it back at higher velocity, which can be captured with some sort of electromagnetic regenerative braking mechanism. Just a toy idea for how energy can be extracted.

It's an established technology

https://en.wikipedia.org/wiki/Gravity_assist

there is nothing speculative about it

Opposite is true also: it's possible to add energy into three body system and increase distance between them (antigraviation). In principle, spherical gradient of gravitation creates possibility to increase orbit using only energy and interaction between masses even in two body system (if smaller body can change it shape and distibution of mass to simulate three body system). I had idea of such aparatus when I was student 30 years ago, but then I forgot the details.

Can't say exactly, but kinetic energy can slightly alter mass. For example, increasing the speed of an object increases its mass.