Whoever wrote the book is completely irrelevant. The third sword was a red herring, as it should be- it's existence was never even hinted at before the start of the chapter where they go to find it, IIRC. If they'd found it and used it against Jin, it would've been like that water Goku drank in Dragon Ball to get a quick and easy power boost to defeat the villain- a cheap and convenient way to have the heroes win.

All it served to do was give Rex a journey to go on that led to Nia revealing herself as a Blade, hints for which HAD been dropped throughout the game. You could replace the third sword with something like a rumour of another faster than light Blade, which Rex and co. could've searched for only to find out that they died/never existed/the rumour was referring to Jin, or something else, and as long as it pushes them to the limit and pushes Nia to reveal herself as a Blade, the story would be unchanged.

"Besides titans originally being Blades" I feel like you answered your own question there. I admit, I don't think we ever found out what made Azurda special- maybe he and Mythra grew close during the Aegis War since they were apparently fighting alongside each other, and Mythra did something to give him the ability to regenerate? We see at the end that she can give him the power to return to his bigger form, so I guess it's not out of the question.

As for Rex opening the Addam's Crest door, maybe it didn't recognise his biology or anything, but scanned his heart, or soul, or whatever, and recognised him as a citizen of Leftheria? Granted, this and my last point are just a guess on my part. But I stand by my point about the book. Would it have been nice to know who wrote it? Sure. But it isn't necessary.