No. Doctors have a saying, "if you are standing on a plain in colorado and you hear hoof beats, dont turn around expecting to see zebras." there are so many other conditions which are more common that i would suspect to cause a cough. Infections like colds and bronchitis when coughing for a week to a month. Asthma or chronic bronchitis when months to years. Others can be found with xrays etc.
No. Common things are common and the most common causes for cough are: 1) post-nasal drip, 2) asthma, 3) gerd. The incidence of pulmonary fibrosis is less than 20 out of 100, 000 people. Shortness of breath morseso than coughing is associated with pulmonary fibrosis. Also, you need to consider what chronic inflammatory processes or environmental agents (i.e., asbestos) you may have been exposed to.
No. Coughing can be a sign of anything, and is not specific to pulmonary fibrosis.