Hi.. Welcome to HEALTHCARE MAGIC.. I have gone through your query and can understand your concerns.. As per your query severe cough along with bad tasting phlegm can occur due to a number of causes, however they can be related to teeth extraction only in case if there have been perforation of the floor of sinuses during tooth extraction and the sinus has got infected.. The floor of maxillary sinus lies close to the roots of the maxillary teeth, therefore, if the sinus floor gets perforated and there is infection in sinuses due to through and through opening with the oral cavity and sinuses the symptoms can be felt.. Other causes can be related to respiratory tract problems like Bronchitis, Chronic Obstructive Pulmonary disease, Pneumonia etc.. So you need to get a thorough evaluation done under supervision of an ENT Specialist and he can advise investigations like complete blood picture, chest x ray, PNS View of sinuses etc to rule out the cause.. Once the cause is confirmed the line of treatment can then only be planned accordingly.. As of now you can take cough expectorant containing Gueifenesin and cough suppressant containing Dextromethorphan.. Hope this helps.. Regards...