You are a Warrior of Light. Not a politician, not a war leader.
You are just here to fight bad stuff and keep Crystal Mom happy. Being naive, clueless and overly passive is a sign of purity of heart.
That's a bit why Midgardsormr seal your connection with Hydaelyn. He wants you to stop being a Pawn of Light and start doing stuff by yourself.
Regarding your character not doing anything while being arrested... well, first, you were weaponless. Then, your powers are still sealed by Midgardsormr. So, no, you are not going to fight an endless flow of guards with your fists only (even if you are a Monk. Punching metal armors barehands hurt a lot).
Besides, your character had no clue what was going on. Risking a diplomatic incident by attacking Teledji and compromising the position of the Scions by doing that would have been a terrible move. Not resisting your arrest was actually the best political move to do.
(I'll also not speak about "you being your avatar" thing because others pointed it out. It's obvious game design.)
Yes, and he admits that himself. He was way to naive and wasn't careful enough.
But, let's be fair here, there's no way he would have known how big this rebelion was. Nobody would.
Speak with others NPC in the world. Especially guild masters and GC leaders. Nobody believes this coup d'etat to be your doing. Besides, Raubahn and Nanamo already knew that Teledji was up to bad things and wanted to take the place of the Sultana by any means. Teledji just showed up and basically said "'sup, I killed her"... Maybe you call that "having a functioning brain", but it was actually very obvious.
Everyone knew that a conspiracy was happening. They just didn't know it would happen that way and that fast, with such magnitude.
"Smirking Dummy" was completly dominating one of the best 1v1 fighter of this world, Raubahn, who was on a complete rampage, kicking stone columns like it was nothing. I think you underestimate the true power of "Smirking Dummy". Y'shtola barely managed to block the blow that killed a lot of people.
Again, regarding your character, you were still weaponless and with your power sealed.
Besides, even if you, the Scions and Raubahn were maybe able to deal with "Smirking Dummy", you would have been surrounded by an endless wave of guards and Crystal Brave. Good job, everyone is now in jail or dead.
Also, since Minfillia can't do sh*t, protecting her (and you) was a priority. You really can't complain about the Scions escape.
Cid comes directly to save you outside Ul'dah, not the Shroud.
That aside, again, speak to NPCs. You'll see that, for example, Limsa Lominsa is only safe because the Admiral will not allow Ul'dah to actively search for you in the city... But you are still considered a fugitive and a Crystal Brave traitor. Limsa, nor Gridania can prevent undercover searches for you... besides, if you would hide in a capital city like that, you would draw a risk of war between said city and Ul'dah, which is now under control of the Monetarists.
The Coertas is the most "neutral" place for you to hide, and the safest (considering the climate and the bad reputation of Ishgard to cooperate).
She's a freaking ninja! Besides, we don't even know how the Au Ra story will fit in.