How does it work with the courriers and maintaining the Veil? Is there some particular way to tell what will satisfy them and help to aid the Veil? The item's level doesn't seem to matter all that much, nor whether an item is limited...

How do you know what to give them?

More limited (and magic) = more better for the Veil.

One way to see just "how limited" something is would be to gain the Legendary Awareness skill via observation experience.

Higher level also helps, but is of less importance than those other two factors.

Is "more magic" related to "more good" as far as Scion despoils and such go? I haven't had a Scion in quite awhile, but was curious.

Something either has the magical flag or not. More limited is better for Scion despoils in general though, yes. But there are a ton of other factors, I've been told. Moreso than the Veil.

Ah, yeah. Too much application of distributive properties. My brain suddenly turned your statement into

More limited (and more magical) = more better for the Veil

I also think I've randomly used what were likely highly limited, and either lower level or just much less useful items on the Veil that shifted it more than I would have expected. Probably stuff that's technically highly limited, but since it's not often used, ALWAYS available. Thanks for clarifying!

Keep in mind that an extra variable here is whether or not your character detects a shift in the veil as a shift in the veil. Generally, being Battle or Nexus is better than being a random guy with detect magic, being higher level is better than lower, and I want to say discern spell helps too.

So if you give an item to a courier/Tahren and it's of middling value, maybe the hero Battle scout registers that as a shift and your midlevel uncaballed transmuter doesn't.

I recently learned the Legendary Awareness skill but don't seem to notice it doing anything. When/how does it work and what does it do?

It's an active skill. You have to use it before it will show you anything.