The male Aes Sedai would shelter in the stedding, but missing the power would always drive them out. The madness was new and unprecedented, and I believe it's stated that they were hoping it would have passed by the time they left the stedding. They were not quite equally involved, it was a gift from one group of male Aes Sedai who had specialised in the study of Portal Stones. The Ways were living things in a way, and the structures inside are formed from themselves. The male Aes Sedai crafted the world of the ways as well as the ter'angreal Talismans for Growing so that the Ogier could grow the connecting Waygates.

I don't have the link for where it was stated, but I believe that Sanderson confirmed that Nakomi was an avatar for the Creator in the same way that Shaidar Haran was an avatar for the Dark One.

I believe that in terms of raw strength, the True Power is stronger and also far more intoxicating than the One Power. Both male and female Forsaken can access it with the Dark One's permission, Rand is able to through his connection to Moridin. Following Tarmon Gai'don the True Power and the Dark One don't disappear, they are simply sealed away as they were before. It was the True Power that led to the Dark One being released, as the Aes Sedai of the Age of Legends sensed it and bored a hole through the pattern to tap it, thus releasing the Dark One. The draw of it was that they sensed an undivided power, and seeing as how the greatest feats of channeling were achieved through men and women working together, they thought this power that was usable by both would be able to achieve even more marvels. The cleansing was accomplished by making a conduit of pure saidar (provided by Nynaeve) and funnelling pure saidin through it. The two ends of the conduit touched saidin itself and Shardar Logoth and Rand was able to use the two different evils to basically cancel each other out. The taint was siphoned off against Shardar Logoth, which resulted in the city being destroyed.

Tel'aran'rhiod is both a dimension and a dream realm. It is a dream realm as it can be touched/entered from dreams, but only by certain people or by odd chance for a very limited time. But it is also a dimension as it can be assessed via a form of gateway. It is a different place of existence, but it is not like the mirror-dimensions accessed via portal stones.

Alviarin was tortured by Mesaana, and after that it was Mesaana who was assaulted by Shaidar Haran. And Galina was captured by the Shaido, it was Liandrin who wound up with the Seanchan after Moghedien left her shielded in Amadicia. But you're spot on with Jaichim. However the thing to note is that the Dark One doesn't directly interact with all darkfriends, only the Forsaken and Padan Fain have been down into Shayol Ghul to speak to him. He does a bit more through Shaidar Haran, but even that Myrdraal tends to only check up on Forsaken. Follows of the Dark One are more predisposed to tearing each other down to bring themselves up, rather than helping each other, so that's why they turn on one another so easily.

Honourable mention for Saerin 🙂

But Mat had made a promise to Rand to see Elayne to Caemlyn, so if they'd managed to dodge him, he would have taken the Band right after her which again puts her in the position of looking like she's taking aid from the Dragon. The girls specifically talk about how dogged Mat is in his promises, so then the Band wouldn't have been around for Egwene to use in her manipulations. Elayne was frequently mentioned as the expert on ter'angrael, so she was needed as the most likely to figure out how to make the Bowl work. They didn't start off with the plan to involve the Sea Folk, so it would have been on her to figure out the workings as the Aes Sedai who also has some experience in Weaving the Winds on a Sea Folk level. Who would you have sent with Nynaeve and why?

But also wasn't Elayne's whole thing that she didn't want to be seen to be 'accepting' the Lion Throne from Rand? Which is what being escorted by the Band (seen by most as Dragonsworn) would have done. And in the end a lot of her support came from Dyelin, who admitted that she wouldn't have supported Elayne if she had accepted the Throne from Rand rather than by taking it in her own right. Even if she tried staking her own claim after arriving, the fact that her arrival would have been facilitated by Rand would have made it harder for people to accept that she was being independent. Not to mention that Nynaeve couldn't reliably channel at the time they were sent to Ebou Dar, so sending her on her own would have been silly. Egwene did not have many true allies at that point, and the Bowl was one of the most important things in the scope of the world at the time, so of course she'd send her two most trusted allies after it.

Do we know if Galad always had the Damodred name, or did he start off as a Mantear? Because with Tigraine being the daughter-heir and presumed next Queen of Andor, I would imagine he should have had his mother's surname, just as Gawyn and Elayne were Trakands. He would never have been in line for the Lion Throne, but he would have been presumed First Prince of the Sword for whatever sister he wound up having, so again you'd want them having the same surname. Was he taken out of House Mantear when Modrellen died and Trakand became the ruling House? I know Taringail always wanted his son on the Sun Throne, but how would that have worked with him being First Prince of the Sword? Or do we just assume there was a deal that the first son would be a Damodred, but the rest of the children Mantears?