“The Diadoch” is what Shayardenes call the heir to the Hegemony regardless of gender. The more proper Karagond version would be Diadochos Phaedros, Diadoche Phaedra, or Diadochon Phaedron.
I haven’t written any heists yet, so I can’t tell you. 
You’d definitely need more than a mugged Theurge’s clothes to get blood out of a bank.
If you specialize in INT, you’ll be as powerful as the average Ennearch by the end of Game 4. But even you plus Sarcifer plus a couple of other equivalent magi wouldn’t be able to take down the Thaumatarch. The “duel” option is just not going to be on the table, I’m afraid. He will always be there at the Siege of Aekos.
Fire-breathing is totally off the table for auto-Plektosis, and wings are pretty questionable. Sorry for all would-be dragons out there.
There will be a bunch of options. You can sell out Nyryal to the Halassurqs, or try to bring it under your own domain, or help it defend its independence, or abandon it and just try to keep any refugees from causing too much trouble on your side of the border.
But evacuating a nation the size of Peru will not be an option. Even evacuating the capital of Nyrnakan, while possible, would be a very tall order. The urban Nyr haven’t been pastoralists for a long time.
The Laconniers don’t have Theurges, and regardless you’re not aware of any Theurgy that would confirm someone’s ancestry. You’ll have the chance in Game 2 to try to figure out how the Laconniers do try to keep track of the key genealogy.
You can definitely attract some defectors by challenging the stifling gender norms of Halassur. But there will always be malcontents willing to defect, and they rarely are enough to tilt the balance of power. You’re not going to be in a place to coopt the whole Halassurq intel network – that’s an overambitious goal for a country at war with your own, when you’re simultaneously trying to overthrow your own country’s government.
In Shayard, the more you break down the social order, the more options you’ll have for promoting social mobility. There was significantly more social mobility in post-1979 Afghanistan than pre-1979, for example.
I’m afraid if you get the whole de Firiac story, they confirm that their sister was killed in a duel.
Any alliance of any kind with Phaedrx will be a deal-breaker for lots of your followers.
Calea would have something caustic to say about the likelihood of this outcome.
The Leaguers are soft nationalists as well as cosmopolitans; many of them like Horion think Shayard would play a special/leading role in the post-Hegemonic koinon. The Laconniers are hard nationalists, angry about the watering down of true Shayardene identity by the invaders’ culture. They will not have the same kind of relationship with a Cosmo MC that the Leaguers can have with a Homelander one.
Not at all – you’re quite right that an MC who leans one way may have good reasons to personally visit the “counter intuitive” territory in G3-5.
Selection by the existing Archimandrites from the upper echelons of the clergy. So yes, a bit like a College of Cardinals, but without any papal election equivalent – the Thaumatarch heads the faith, most definitely without elections.
There’s nothing resembling a “thieves’ guild,” but some forms of organized crime go on among both drudges and merchants, and you’ll be able to engage with that in Game 2. (As well as with smugglers and pirates if you spend time with the foreigners.)