When Detective Logan and Detective Wheeler are questioning the doctor who wrote the prescription for the Methadone, the doctor said he called it in to a pharmacy. Since Methadone is a Schedule II narcotic, it cannot be called into a pharmacy, the patient would need to bring in the hard copy of the prescription and it has to be written on a security blank.
While interviewing Lorelai's doctor he said that he hadn't examined her in three years and that he called in a prescription for methadone to her pharmacy, here he violated at least one, possibly two regulations for prescribing controlled substances. Since methadone is a schedule II controlled substance a pharmacy cannot dispense it without having the original hard copy of the prescription, a doctor can't just call in a verbal prescription, or fax one, like with drugs in lower schedules. However a doctor is allowed to call in a verbal prescription, or fax a copy of the original prescription for a schedule II drug to the pharmacy so that they may prepare the prescription ahead of time and have it ready for pickup by the time the patient arrives at the pharmacy, however they would have to get the original hard copy of the prescription from the patient at that time before being allowed to dispense the medication. It is possible this is what Lorelai's doctor did since she was very impatient and seems to be the kind of person would would balk at having to wait at the pharmacy for the prescription to be filled. However he did violate regulations for sure when he wrote Lorelai a prescription for a schedule II controlled substance without examining her. DEA regulations state that a doctor cannot issue a prescription for a schedule II controlled substance if it has been more than 90 days since they last examined the patient, and they cannot issue new prescriptions for schedule III, IV & V controlled substances if it has been more than 180 days since their last examination.
Lorelai's doctor says he started her on methadone because it was safe to use during pregnancy, which meant she was dependent on another opioid at the time. Methadone is actually in pregnancy category C, meaning that it carries a moderate risk of causing harm during pregnancy and is ill-advised. Almost all opiates are category C so methadone isn't any safer than any other opioid. Women are strongly advised to stop taking any type of opioid as soon as possible upon learning they are pregnant, usually they are weaned off opioids over the course of a few weeks. Not only can opioids affect the development of the child if the mother is taking opioids in the latter stages of pregnancy the fetus will become dependent on them as well and will go into opioid withdrawal soon after birth. Women should also not use opioids while breast feeding because opioids are secreted in breast milk. Lorelai's doctor was being highly irresponsible by continuing to prescribe her opioids during the final moths of her pregnancy, if she refused to wean herself off them then he would have been ethically obligated to try and get her into a drug treatment facility, preferably one that specializes in treating pregnant women. While he couldn't force her to stop using opioids he could, and should, have told her that he would not continue to prescribe her methadone or any opioid while she was pregnant.
When Logan asks Rodgers, "accident, suicide, or homicide[?]," she responds, "case undetermined pending police investigation." It's an acronym they've used before, CUPPI, but it's always been "cause" undetermined, not "case."