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.