"Just" a plot contrivance? What does that even mean?

No, but had he attempted to Heal a man, it would have had a similar result to Nynaeve Healing the women. IIRC, RJ even confirmed that a man has to Heal a woman and vice versa for them to recover their full former strength. It ties in with his themes of interdependence of the genders.

And those are also fairly obvious misdirection tactics to tell people he didn't do it. You can't trust things a character says out loud, because what he says is not really what is true or what he thinks is true. Most of the time, characters say what they want other people to believe. Only his own self-directed thoughts or dialogue can be trusted, and even then, you have to beware the PoV trap.

Well, since this book outright states that Katerine is Black, that's not much of a guess. By the same token, it's kind of a stretch to lump Nesune in with her. Their meetings with the nobles are not at the behest of the Shadow but the Tower. They might also be advancing the Shadow's plans, but that is coincidental, like the women who advanced the Shadow's plans by voting for Egwene. It doesn't make them Darkfriends, it is merely one more example of how the Tower ends up doing a lot of the Shadow's work for it. Anyway, we see the Salidar embassy doing much the same thing in Caemlyn as the Tar Valon does in Cairhien. This was likely written in such a way as to affirm the similarity between both groups, regardless of their political disagreements and support the portrayal of the rebels as antagonists to Rand to PoV-entrapped readers who might have whole-heartedly embraced the cause of the rebels and now see them as all good and the Tower's as all bad.

Further, the meetings with the nobles are part of the efforts of the Aes Sedai to control Rand and the rest of the world. They want Rand firmly under their control so they want him separated from any sources of power that he might use to resist them. If the nobles in the countries he rules support or listen to the Aes Sedai, then Rand has less control over that country and needs the help of the sisters to actually get things done there. They are also trying to pave the way for getting things back to normal. That's why in later books there will be so much consternation among sisters over his appointment of Stewards in Illian, Tear and Cairhien. These new offices change the playing field and are completely outside the old ways of doing things, which means the Aes Sedai have no strings with which to manipulate them. Most of the fuss and bother among Aes Sedai over Rand's political actions stems not from what he's actually doing, but that he's making things on his own, and every new thing he does is another new thing over which the Tower has no influence. That's what was behind the things Dobraine and Berelain were advised by the sisters - in each case, the sisters wanted them to go back to their old places on the gameboard, to reset to the status quo. It's not a Shadow thing, it's a Tower thing (though it may serve the Shadow's ends as well).