I guess for the sake of argument, what happens in Wheel of Time is that rather than fully die, people are stuck in a post-physical state of existence that keeps making them born again and again. The soul has to do with your will, your essential nature and no one and nothing can change that but you. That would be why people keep being reborn, and the Dark One can't do anything about it. If people are bad enough or closely aligned with the Shadow, he can mess with them for a bit, but eventually, they die to that state of existence as well, and go back into the pool.

As far as non-human constructs go, those that have souls (I actually believe it's stated somewhere that Myrdraal do not) did not have human beings using the Power to put souls into them, rather, they are somehow derived from natural human life. Trollocs, for example, were bred from humans, which is where they get their souls from. And humans cannot duplicate the Creator's work 100%, which is why in certain conditions, they fail, dying when passing through a gateway being the most notable, so clearly they are not actually all the way in control or successful in the process, rather they're just hijacking the natural mechanism through a physical process. It's like a test-tube baby - no one is actually making new people or creating life, they are simulating the conditions under which life begins, so nature takes its course. Making Nym or Trollocs or whatever is simply a more complicated elaboration of that.

Likewise, Tel'Aran'Rhiod is just a different state of existence. Their bodies may have stopped functioning, but the Heroes still live, on this side of that ultimate door between life and death. What one sees in Tel'Aran'Rhiod is a real thing, like the emotions and feelings an Aes Sedai and Warder sense through their bond, but it's still not the person's soul and a Hero of the Horn is just not all the way dead yet. Death is nudity, the soul is the naked person, the spirit one sees in the World of Dreams is clothing and the body that dies in the world is a suit of armor. The Heroes of the Horn or the Forsaken never get all the way undressed, the Dark One or the Pattern intervenes and they get a new suit of armor. That's what balefire does. It destroys your armor, but it can't kill YOU. It rolls back time so the Dark One doesn't have time to nab you before you get all the way naked, or maybe in the process of rolling back time, it also destroys your clothes. Either way, you are still there, you're just going skinny dipping or whatever happens after you die, for which we have no frame of reference, being limited to perceiving only suits of armor. When people see ghosts later in the books, they are seeing clothing, whether or not the clothes have people in them or are just moving around in imitation of what their wearer used to do (in the Dresden Files, for example, it is the latter).

I don't know how things work in WoT, whether you just keep changing armor or get all the way naked and put on new clothing AND new armor when you are reborn (I would assume so from things Jordan has said, and also that would be how the Heroes are different - they retain the clothing from life to life), but the soul HAS to be something more, something not truly of this world, or else nothing really does matter. The point of being good even when you fail in life or in the world, is that you are being true to that part that is you, regardless of the clothing, and that compromising morals or values to prolong existence is like living for your clothes or armor. The One Power or the Dark One can hijack your armor or move your clothes against the will of your limbs, but they can't change your will. Only you can do that.

Coming back around to the Gray Man question, I think they vacate their armor for the Dark One to use. That's why the wolves call them Not-Dead. They are dead, but don't act like it. I don't think RJ ever portrays a Grey Man acting with a will or choices (because they don't have those), just reacting to circumstances. But because of their cooperation with the process, they don't look like traditional undead. Really it seems like all the categories of humanoid Shadow minions have an aspect of the vampire mythos. The gholam drink blood and can pass through cracks and walls but can be stopped by certain objects worn about the neck. The draghkar fly like bats and drain life, the Myrdraal have the superhuman strength and pale skin and freaky interactions with reality vis a vis shadows, mirrors and reflections, and can't cross running water. For the Gray Men, it's the walking corpse aspect.