To the Cortana question: According to all the books (I've read them all lol) rampancy is something that cannot be cured and is a symptom of basically too much thinking and processing power. It occurs after about 8 years and is caused by A. I. thinking themselves to death. It's like if you could think 10000000x faster and you never slept. Your life would be cut short dramatically. So knowing this I don't think Halsey would have been able to "cure" Cortana. John was just desperate and looking for any straw of hope to grasp on to. I believe that deep down, MC knew Halsey couldn't fix Cortana (the guy is a genius after all). This isn't too say that Halsey absolutely couldn't, but I highly doubt it given what the books say of rampancy and no "smart" A. I. has ever been fixed of it. Cortana is actually just a younger Halsey. Halsey cloned her brain and created Cortana from it so she not only thinks like a young Halsey, but her entire physical form and personality are a young Halsey. Considering the fact that Spartan 2's were basically raised by Halsey and see her as mother, it's no surprise that MC in his desperation held on to the false hope of Halsey fixing Cortana. He used this false hope to motivate him. But in reality, it was most likely a lost cause.

To your last question about the great journey: I believe it was the San'Shyum (the Prophets) who first discovered Forerunner technology on their home world as well as a few other planets and in doing so they discovered evidence of the "great journey" and misinterpreted this evidence as some kind of divine transcendence. They then used the advanced Forerunner technology to defeat and assimilate the other alien races that make up the covenant. The great journey was then taught to all the races of the covenant as they were assimilated and was used to help unify them. Not all the prophets believed that activating the Halos would trigger a divine exodus, some of them were starting to think that the Halos were weapons of destruction, but it was heresy to say this so they kept quiet. If you read the book Halo: Broken Circle, one of the main characters whom is a prophet starts to believe this. Halo: Broken Circle is definitely the book you want to read if you want more info on the origins of the covenant and the great journey. The book is basically about how the prophets fought, subdued, and assimilated the Elites, and the effects that would have later. And you won't get a better source of info on the San'Shyum as a people and the beginning of the covenant as a whole. Really good read if you are a fan of halo lore.

Lol, thanks :) Looking in hindsight I probably should've made separate replies, although I just kept thinking of stuff in my head and adding on to it...

Hunt the truth season 2 was just announced