The only thing we can go by in the story is the explanation given by Moiraine in EotW. It's certainly possible that she was mistaken, but she is nevertheless clear in that explanation. According to the history she related, Mashadar was an evil that was created by men in their hatred of the Shadow during the Trolloc Wars. It attacks Shadowspawn even more vigorously than any other living things, which is why the trollocs chasing the Emond's Fielders had to be driven into Shadar Logoth. It also couldn't leave the city until Fain enabled it to much later. It would be odd for this preference in victims to have developed before even Aginor created them. If he knew about it, it would still have been something new for him. Not ancient. It could be that, like Machin Shin, Mashadar was born of the Taint. But its antagonism to Shadowspawn and its being tied to Shadar Logoth both strongly imply that Moiraine was right about when and why it was created. And again, the Taint was brand new for the just-released Forsaken. They had either not seen any of this until just before they were trapped, or not until after they were released. There are a number of things (like a'dams or the Warder Bond) that are old to the people living at the end of the Third Age, but which the Forsaken have never heard of. I believe Mashadar is another one.

RJ introduced a number of rules into the mechanics of the One Power that are almost geometric in their nature. For most things, the comparative strength of channelers matters. But when you get to 13, the "geometry" outweighs that. A circle of the thirteen weakest female channelers can shield the strongest possible male channeler, in effect without breaking a sweat. It takes fewer channelers to maintain a shield once formed than to form it in the first place, but the same kind of "geometry" still comes into play. Six can maintain a shield (as long as they are doing so actively, rather than tying it off) due to the way they are spaced around the shield because they can react as a group to counter anything the victim attempts. So thirteen is the number for turning because that's the number where strength in the Power no longer matters. Logain was unable to prevent being shielded when he was captured because his strength in the Power was irrelevant. He resists turning not because of his strength in the Power, but because of his strength of will.

Because they weren't "out" yet. Except for Ishamael, all the Forsaken were still trapped in the Bore until the Seals started failing - extremely recently in the story. As far as why they couldn't have just tracked Someshta? It's because (by design) he wasn't actually in the world unless and until someone's need called him. This was the first time since any of the Forsaken got out that the Green Man was touching the world in a way that made him detectable to them. Remember that Mashadar was created more than a thousand years AFTER the Forsaken were trapped. It would have been a new thing to them, not something ancient. Only the things at the Eye would have been ancient to them.

No, you're right. My mistake. The scene you're referring to takes place in Path of Daggers, chapter 26 "The Extra Bit." A few different Sisters channel into the Oath Rod and re-swear the oath not to lie, to prove that they are telling the truth. So far as I know, it's the only place we see that. But others in the same scene are done as in the Raising ceremony, with one activating the Rod, and another swearing. It appears that it's more tradition and ritual, rather than necessity. --- I was looking for that scene before I posted, but couldn't remember where it was. I should have waited. 😌

I think you've got the timing a bit wrong. What leads to Egwene's capture is her concern over Rand's wellbeing - played upon by Liandrin. The Girls don't even know the Black Ajah exists at that time. They aren't set to hunt them until they get back to the White Tower following her rescue and the Battle of Falme. None of the Little Tower actually "forget about" the possibility of the Black Ajah either back in Tar Valon or with them in Salidar. Rather, they actively avoid talking about them. Partly because the possibility terrifies them completely. Those are not the same thing.

If she could have freed herself with the Oath Rod, she could have done so as soon as she got her hands on it. She wouldn't have needed to escape in order to use it. But she couldn't. As was already said, she remained bound by it not to channel without permission. But even if she could, she still needed someone else to channel into the Rod in order to release her. I believe that in every example seem in the books, the person swearing an oath is never the same person who is channeling Spirit into it. It always takes two channelers - one to activate the Rod, and another to swear the oath while touching it. Even with the Rod, she couldn't release herself. She needed another channeler.

It's not just that she has been taught to lead. She has been taught to recognize who has the right to lead in any particular situation. That means that before she left Caemlyn, she knew to follow her mother, or Gareth Bryne, or Elaida, depending on the circumstances. When she got to Tar Valon, she knew to follow the Amyrlin, or any other full Aes Sedai. On the road with Egwene and Nynaeve, she could lead or follow as the need arose, but (mostly) always with diplomacy. Elayne almost instinctively understood the chain of command, and that chaos was the result if it was ignored.