How did he get to Mythal then to take her power? There would have had to have been at least one active eluvian he had access to, and not Morrigan's as he had already left Skyhold. It makes sense to me narratively, and makes the story more interesting to me, to think that he was doing his own thing behind our backs at the ball while we were running around trying to fix Orlais.
Opening one eluvian likely isn't the same as opening a full network. We see Kieran open one after all, and I don't think he was in command of entire network.
And I'm kinda on the opposite side... like - there was so much going around and Briala and Inquisitor were wrapped up on the same thing that Solas doing something was kinda out of question, especially if he was in the core team that went with Inky to roam the Palace. For me it seems unlikely that Solas went after Briala - in fact I don't think there was any need to go after Briala. Especially that - like I said - he didn't say that he took the key from her. He simply said that he had to override the magic - something he'd be able to do when buffed with Mythal's powers.
Wasn't it already established that she sent her wisp through to preserve it for Morrigan? It looked like what she took from Kieran (though obvs not everyone had Kieran- and, omg, there's another theory I'm loving right now - that there weren't two wisps sent through the eluvian if she took Kieran's, because all of the Old Gods have pieces of Mythal split up into them. So, basically, she was reclaiming a piece of her own spirit!) and if both Solas and Mythal already had the ability to access eluvians there would be no need for her to waste power on reactivating the network if she knew he could do it. Even the Qunari who hate magic and had to figure them out on their own can take control of eluvians, I don't see why Solas who is far more knowledgeable about them would have a difficult time.
Um, no - that's an assumption made by players. A rather popular one, but still an assumption. Not even the dev notes say anything about it (and they can be outdated/inaccurate anyway), with them only mentioning that eventually she wants to bestow her powers to Morrigan = although how she does it is anyone's guess. I assume however that she locked enough of pieces of herself throughout years in various horcruxes artifacts, considering that she's a careful planner who had years to put stiff into motion, so I don't see how she'd need to push the part of herself right before she gets killed, like it's something she had to do in a hurry.
Also - the Qunari apparently still had to have keys to open each eluvian and only had access to select ones they managed to crack. Solas apparently is powerful enough to control the who;e network/section, though apparently Qunari keys still work...