I'm a nurse--not in an A&E department, but still in a hospital.
a) The Machine That Goes Beep. There are actually a lot of machines that go beep. The two that are used most commonly on our ward are an oxygen saturation monitor (generally used intermittently for spot checks, but running continuously for patients with the potential to develop airway or breathing difficulties, on our ward generally post-tonsillectomy, or anyone who needs a "close eye" but isn't critically ill), and a cardiac monitor (running continuously). High dependency patients, including patients in diabetic ketoacidosis, patients who have had treatment for a paracetamol overdose, and patients struggling to recover from an anaesthetic are all likely to be on a cardiac monitor. Whether or not the machine actually goes beep or not depends on the brand, & the settings required for that patient. I'm guessing Faye is on a cardiac monitor--I don't know much about the treatment for alcohol poisoning as it's not an issue we see often on our ward, but if it's anything like the treatment for paracetamol overdose, the treatment itself can cause cardiac issues, in addition to the original poisoning.
b) Marten not being in the last panel. Faye is looking at the hand on the side of the bed with plugs & sockets & machines. The emergency equipment on our ward, and every ward I've worked on, is on this side. If a patient as ill as Faye was in my care, I would want the friends/relatives on the other side, to allow for easy access to the oxygen/suction/emergency buzzer. It may sound stupid, but the thirty seconds it takes to get someone out of the way in an emergency can make a massive difference--for example, if Faye started choking on her own vomit and the nurse couldn't get to the suction to clear her airway. It's also possible that she was being rushed to the ICU in the previous panels, whereas she's now arrived, and the staff have asked Marten to leave whilst they get her settled/have handover from the paramedics to the unit staff/possibly change her out of vomit-stained clothes & other things that she might not want him to see if she was entirely lucid. People whose friends do really dangerous things (like get unconscious-drunk) due to mental health crises are often angry, but more about the damage they've caused to themselves than anything they've done as a result of being in such a mess. I seriously doubt that one more punch from an obviously-impaired Faye would make someone as caring as Marten bolt.
Also, excellent/heartbreaking comic today. Those final two panels are chilling (although I agree that she is not dead, just drifting out of consciousness).