Hooo, boy. That was fun.
Regarding Detective Authority being plot armor - not sure. Erika did said
many times that she won't die because she is a detective. While not in red, that many times is enough to make some conclusions.
Detective's Autority itself wasn't confirmed in red, Battler however did
confirmed it in white (as a piece), and Erika was
confirmed in red as a detective by Beatrice.
Conclusion: eh. Erika is without a doubt detective. Also it seems that piece Erika knows that she is a
part of a gameboard (despite her meta version not being present anywhere near the story; and no, I don't believe that there's some other version to this "gameboard"), so we maybe probably can believe her words regarding the detective's "immortality" to some extent. Not sure.
For abilities:
Social Influencing for Erika. Yes, that's all -_-
As a side notes:
maybe we can add Purple Declaration to the Truths page, and give it to Bern.
Erika's intelligence listed as "Very intelligent". Why not Genius is a real mystery to me (I may even argue for Extraordinary Genius, but eh, another time).
That's all for now. I thought that I can find a bit more in this manga, oh well.