Do anxiety can make make coughing?

Cough. Usually not but some people with anxiety feel a sensation that there is something in their throat that they constantly need to clear away by coughing. It the cough doesn't get better you should see your doctor to determine the cause and possibly get a chest xray given your smoking history.
Yes. Anxiety can cause many symptoms and signs including cough. Anxiety can provoke hyperventilation, and for patients with history of asthma or copd, it can cause flare-up of these disease and worsens wheezing, cough or shortness of breath.