The twins had RSV, or respiratory syncicial virus, which is not well publicized but can be a very bad virus. It can lead to respiratory failure, as evidenced in one of the babies, and treatment is mostly supportive, including intubation and a respirator.
Scott does not have a bone marrow match, so he's getting an experimental chemotherapy. I'm assuming he's getting some high dose combination of drugs. I do think they're stretching credibility by having him look OK during the day, get chemo and immediately develop near fatal sepsis, and then to begin antibiotics and feel better again.
I also think Carol made a mistake by allowing the Carter Foundation to be her only source of funding for the clinic. At $150,000 minimum that's been donated from just one source, she should have multiple sources just in case one is reduced or eliminated.