One issue is that at the end of 2.0, during the Praetorium fight, the only thing Gaius has changed his mind about was that he, personally, was the provably correct choice to rule Eorzea.
Instead, after having been defeated by us (three times: as himself, in the regular Ultima Weapon, and in the Sabik-unleashed Ultima Weapon), he goes "well, maybe it's true that you're stronger than me, in which case you must lead Eorzea."
At which point I really, really wanted my character to go "Gaius van Baelsar, you still don't get it. Being strong doesn't confer ability to rule. That's not how this works. That's not how any of this works."
So when we meet him again and he goes "I was wrong to think that the strong must rule", it's already a major change in innate philosophy that he hadn't displayed before, and indeed showed no signs of changing previously.
By "Resistance Reports" they mean the "Resistance Officer's Reports", ie the Ala Mhigan Resistance. It's the interactable pile of papers at Porta Praetoria in the Lochs.
The "Gaius may betray us" line is in Gaius's bio as written by the aforementioned Resistance Officer.
