FOXDIE was Naomi's invention from MGS1. She made it and programmed it. Snake was injected it before the Shadow Moses mission. It's been lingering in his body since then. In Act 2, Naomi tells Snake that the FOXDIE was mutating, making it possible to kill anyone once it was completely mutated. The syringes had nanomachine/FOXDIE suppressors so that the nanomachines would shut off since they reacted badly for Snake because they were connected to the System. That is also why Naomi began to die from her cancer since the nanomachines were keeping them in check, and Vamp became "mortal." I'm also pretty sure their nanomachines aren't connected to the System since you never see them react whenever Liquid to control of the System. Drebin also injected you with the new FOXDIE so that you could use the blank ID guns, which plays a small role at the end. Big Boss explains at the end that new FOXDIE was uprooting the old, so Snake was no longer going to produce a deadly strand of mutated FOXDIE. However, he did imply the new FOXDIE was going to mutate eventually, but obviously not in Snake's lifetime and probably not that all an important a detail.
The new FOXDIE killed the founding members of the Patriots, not the old FOXDIE.
As for watching the last cutscene, it should be on YouTube.
To summarize, Major Zero created the Patriot AI's to keep the Patriots' ideas alive forever. Unfortunately, some of the AI went haywire and basically made it so that war exists to make money. Hence destroying the AI freed the world of the Patriots. Basically the AI went rampant a la Halo.