What are potential illnesses of these symptoms, hot and cold flushes and coughing fits?

Several: Hot and cold flashes could be from thyroid problems or other hormonal imbalances. Cough could be asthma, bronchitis, copd, pneumonia or even reflux. Patient who have both are often suffering a lung infection like bronchitis or pneumonia. You should get this checked so that a proper diagnosis can be made and treatment rendered.

Infection.: With those symptoms alone, one good possibility may be some kind of respiratory infection. These can be either viral or bacterial, so treatment may be needed.