You didn't have breathing problems when you asked for an appointment; you had a cough. Then when you had breathing problems, you sought out emergency care and had that taken care of. But you blame your original doctor for... not predicting your breathing problems? Or not giving you lung x-rays the moment you diagnosed yourself with pneumonia? I'm confused what you would rather they have done.
The missed allergy sounds like a legit failure, though, but I'm not a doctor.