Veteran Rewards - Clarification on mission rewards

I have a couple of questions about the mission rewards that can spawn in the Veteran Rewards Vendor.

Of what I have read, I’m assuming those are per character, not per account. But what about normal mode/tvhm? If I have done a mission on normal, can that reward spawn on tvhm?

And how are things in coop? Do the host’s missions count, or my own? Is it different in cooperation/coopetition mode?

Any character will see them in the machine (I have several, and I hit them all with each regularly). Have we confirmed that one has to have completed the missions for the gear to show up? It doesn’t seem that way.

To me it very much does seem that way. My characters that have done pretty much no sidequests keep seeing the same few items over and over. Like Traitor’s Death, Cold Sholder etc. As legendaries go, I have seen a number of Red Suits, but no Cloud Kill yet.

I check this machine constantly - have seen almost everything (I think), but never seen a Cloud Kill (or a Rogue Sight, which is actually what I’m looking for most at the moment).