Well, others who haven't missed paragraphs showing Egwene as a sensible and stable person ridicule those worries precisely because the books make it clear that Egwene is the last person Rand needs to worry about. It has been made so clear that Egwene understands what Rand is going through that ignoring it and saying she'll go back to bickering with him as they did when they were kids is absurd.

No it does not. She was cold and professional because it was a professional issue. She cannot look at it as a matter of friendship or love.
Which is precisely why people like Siuan respect her so much and work so hard to keep her Amyrlin. From Siuan's PoV, we know that Egwene is more casual and friendly around Siuan. They are not just successful as mentor and student, but as friends. And Siuan respects the fact that Egwene cannot always treat her as a friend. She couldn't be a good leader if she did.
It is so immature and childish to expect a woman in Egwene's position to act cuddly and sweet when her subordinates disobey her. Its like a kid thinking their teacher hates them because the teacher punished them for their transgressions.

Who said flee the Forsaken? Semirhage was shielded. What if they took her and Rand away from the scene before Rand came to and had a chance to speak to the sul'dam?
Rand would have gone ballistic. He'd have had a right to be angry, but he'd have mishandled it badly. He'd have yelled. He would have lost respect.

Yes, I was wondering why you mentioned that, since Rand told her he's going to use her husband's death, and threatened her with the Choedan Kal!

She seems less damaged because she trained herself to overcome the damage. Or do you think two months as damane, when punishment involved being made to feel like your submerged in boiling water, is a walk in the park?
Rand has never been captured and mistreated for so long a time. He has never lost control over his ability for so long a time. His other issues make him have a worse time, but a lot of them are because unlike Egwene, he has refused to bend to the reality of his situation and make the best of it.

But we have no idea that Bloodknives were still there. they may have been killed by the Warders, for all we know. And her loyal Novices would have moved her anyway! Egwene is right that no matter how worried Siuan felt, she shouldn't have let that be the basis fo her decision to kidnap Egwene.

So? Since when did it become okay to be an ass if you were under stress? Rand may face greater stresses, but her also has greater power at his command. Egwene would never have floundered about like Rand did.
Understand that I'm not saying Rand doesn't have reasons. I'm only saying that those reasons can serve to help us understand, but not excuse, his behavior.

Rand has had about as easy a time when he's tried to impress anyone. He doesn't try, most of the time, that's all.

She irks you because you're determined to be irked, and because you conveniently forget situations where she shows how sensible and mature she is. There's absolutely nothing wrong with her attitude. Rand needs to learn from her.