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.