welcome to HCM
Sorry to hear about condition of your father.
In avanced lung cancer
, there can be bleeding from tumor into the airways. That can be the cause of red mucus in his cough.
means he has bleeding some where in his G.I. tract.Sometimes patient having blleding in airway can swallow it and develop black stool.
You mentioned your father recently received chemotherapy
an sometimes occur after chemotherapy if blood counts are very low. You should inform your doctor regarding these symptoms.