Hi.. Can understand your concern.. As per your complain in case if you have a deeply infected tooth as well as if only root pieces of tooth are left they are a potent source of infection leading to pus formation below the root tips of the infected tooth known as a DENTAL ABSCESS which swells up the gum and oozes pus and blood.. You need to get them extracted as until they ill be present infection will keep on sprouting and cause pain and pus discharge.. I would suggest you to consult an Oral surgeon and get evaluated for the cause of not getting numbness with local anaesthesia and can apply other techniques for numbing the teeth and the soft tissues other than the regular one to numb directly near the nerve trunk or can remove teeth under general anaesthesia (if local does not work at at all). In case if there is generalized bleeding from the gums then it can be due to Gingivitis which can be due to infection and inflammation of the gums leading to bleeding and swelling of gums.. For this you should get the scaling of teeth done followed by using an antimicrobial mouth wash and brushing teeth twice daily and flossing can help.. Only using mouthwash cannot help and can have an additive effect once the cleaning of teeth is done.. Hope this information helps.. Thanks and regards.. Dr. Honey Nandwani Arora..