Well, that very nearly is the method of power generation. It would presumably be paired with some sort of Dyson sphere around the white dwarf. (Remember that a "Dyson sphere" isn't necessarily a solid shell, but just some configuration of matter where no matter which direction the energy goes it is intercepted by something.)

You may also enjoy: https://www.youtube.com/watch?v=t-O-Qdh7VvQ which is a minutephysics on black hole power using much the same principles but attaining yet higher efficiencies.

So we would decelerate the trash, it hits the white dwarf, gets crushed, and then undergoes various nuclear transformations resulting in energy output we would then capture with our Dyson swarm?

Sorry, but it has to be rigid to be a sphere. Angular momentum is a bitch!

PS: Dyson Spheres is misnomer, Olaf Stapledon should be credited for this idea (Star Maker 1937)

That's more about why "sphere" is a misnomer that leads people to think the idea requires a large rigid shell than a problem with the idea. You can create configurations of matter that continuously and completely englobe a star without it being a rigid sphere. What you can't do is make it so that any other configuration is always receiving sunlight at all times because no part of it ever in shadow, but that is not a requirement to capture all energy from a star, that's merely an engineering question around efficiency.

The result may not be literally a rigid sphere as people think of it, but if the orbits are packed tightly together enough it'll look plenty "sphere-like" from the next star system over.

A rigid Dyson sphere is probably impossible, and its utility is not entirely clear. Making a Dyson swarm around a star is merely a lot of work, and also has the advantage of immediately paying benefits as soon as the very first satellite is launched.

I've read somewhere that we now have a ton of junk satellites orbiting earth, and if they happen to crash into each other it could set off a chain reaction that creates a satellite-debris shell around the entire earth, roughly resembling a non-rigid sphere. Is that true?

That's roughly the story in the movie Gravity.

Collisions between orbital junk are most likely to happen between objects in the same orbit. And most junk is in some roughly equatorial orbit. So I'd expect the resulting debris to end up in roughly the same orbit, so that the debris would form a sort of fuzzy disk - not a sphere. I wouldn't expect to observe much debris over the North Pole.

Is "damping" a technical term, distinct from "dumping"? The article uses it more than once so it seems deliberate...

Damping is generally a technique which reduces, slows, or restricts a system. It is not a synonym for dumping. We have old, related terms like the "damper" on a fireplace, which I suspect may invoke a more basic analogy of putting water on wood fuel, i.e. making it damp and slower to burn.

In modern usage, "damping" usually refers to a drag or parasitic affect in some oscillatory system, but it does not have to be oscillatory. It can be other drag effects that convert energy, with a usual assumption that it will bring the system to a new equilibrium. An object falling through the atmosphere can be considered to have its fall dampened by the air drag, converging towards its terminal velocity.

I am not familiar with all the physics involved in falling towards/into such a star, so I do not know if there are field effects which would cause damping of the fall prior to impact of the trash with an actual surface, similar to the atmospheric descent towards earth. I also wonder if they imagine the orbital system is being damped to allow the trash to impact the star, by bleeding off tangential velocity to decay the orbit.

Damping is a physics or maths term. To damp means to (steadily?) decrease the amplitude of a wave or oscillation.