Ferane wasn't asking for advice, she was testing Egwene, as Egwene herself pointed out. Their only difference was on whether Rand had to be controlled.
As for the Blacks, the only reason Egwene gained control is that she made them uncomfortable, which she was able to do because she figured out they had illegally used a fourth oath compulsion on Meidani. That may well rank as a stilling offense, and if she had blabbed, their BA hunting days would have been over.
Even after that, Saerin never shows any undue respect till Egwene straight out asks her if she wanted to be in charge. Saerin says she does not, so she ends up supporting Egwene.

But we see worse in other fantasies. Danaerys in aSoIaF turns from timid teen to commanding queen in a matter of days, and no one objects.
Egwene, however, was always driven, and has been shown to be a perfectionist since she was nine years old. She's had months of intensive training from a bunch of very intelligent and forceful women (Moiraine, the Wise Ones, Siuan, etc.), and has always been shown as a sponge for knowledge, and a person who adapted to new environments very well.
It isn't like there is no basis for Egwene's competence.

But Elaida's declining sanity was always being built up, and the foolishness of some of her actions like kidnapping Rand or demoting Aes Sedai are not things that were newly discovered now! People on this board have been finding the same flaws with them for years! Egwene merely pointed the illogic of Elaida's actions. She called her out on the whole fourth oath thing, and even that was not without precedent. Elaida was dreaming of it in book 7, I think!
As for the BA hunters, I explained why Egwene was able to take control. Nothing surprising. She was subtly blackmailing them, just like Nynaeve blackmailed Siuan when she found out she and Leanne were not enemies. Doesn't make them idiots.

Not necessarily. Tuon had planned the attack before she talked to Rand, and had always seen it as a bargaining chip. Sure, a truce may have prevented that, but Egwene did not need the Seanchan attack to become Amyrlin.
All she had to do was approach Saerin with Verin's book, promise swift retribution against the Blacks, and she'd have had four powerful Sitters in the Hall loyal to her. Suana and Ferane were also favorable to her, and they being Ajah Heads, could have brought in more.
Elaida had already lost face, and Egwene would have played the removal of Silviana to turn the Red against Elaida, while making them see her as a better candidate.
It would have taken a little longer, but the Seanchan attack wasn't necessary.

And we also know she is incapable of being manipulated by them? Egwene says somewhere that Elaida had begun to rely on Katerine increasingly after Tarna left, a clear sign as any that the BA was pulling Elaida's strings.
[quote]One might expect a little more intelligence and smarter moves from the Black Ajah than from Elaida. They could've killed or abducted her, thus not only removing the annoyance inside the Tower but also further diminishing the chances of the two factions reuniting. With the numbers the Black Ajah now turns out to have, you'd think they could've done more than just follow Elaida's orders.[/quote]
But killing or abducting her had grave risks. She was constantly watched, for one, and her death would cause immense turmoil within the Tower itself. Elaida would not be accused of it, and thinking sisters who investigated might have found out more than the Blacks wanted.
Instead, they got Katerine close to Elaida, and as MoN, Katerine was to have control of Egwene after she was released. Do you doubt they wouldn't have tried to reverse her actions once Katerine had taken care of her? (Of course, they need to be forgiven for not knowing that Katerine would likely be less effective than Silviana).

Why was no leadership required. Please explain how, with all the divisions and infighting in the Tower, anyone like Seaine or Saerin could have provided leadership.

A month where a lot happened, shaking the Aes Sedai's belief in their infallibility.

But she is an enormously strong woman (in the One Power). That does count for quite a lot, you know. Her strength and skill had even Elaida respect ehr enough that she did not want to kill her. And silly though it is, OP strength does play a subtle but ever present role in the way AS look at you.

But they still haven't faced large scale battles against channelers in forever. And they were sleeping. And did not know what the Seanchan could do, did not know about the collaring, etc.

Who says they didn't? But it is uncertain how many of each Ajah were present, since all but the Red's were split. Worse, the Yellows and Browns had their quarters split between their old locations and the Novices quarters and the basement kitchens! Add to that the fact that many rooms are likely to be empty, it is very possible that the AS were not concentrated enough to orm large circles. They thus got taken down in small groups. Hardly an impossibility.
Egwene had the advantage that a lot of novices were clustered together around her.

Not as convenient as the Choedan Kal's Access Key shrinking from a three foot long statue to a something Rand could carry in his pockets! I don't see too many people complaining, though.