I have had a persistent cough that has got progressively worse, I have hot flushes and also pain and discomfort in back at bottom of rib cage.?

Possibilities... The possibilities for a cough of over 3 weeks duration are bronchospasm from any etiology, gerd, post-nasal drip or a combination of these factors. The pain in the rib cage is probably musculoskeletal from the cough. See your doctor for diagnosis and treatment as all of these have different treatments.
7-10. That's how many days from the onset of symptoms you may push your luck trying to let it get better on its own. You have inflammatory symptoms suggestive of pneumonia/bronchitis with pleurisy. Pleade seek advice.
Depends. Chronic cough can be from many things - post nasal drip, asthma, allergies, medications, infection, even reflux (heartburn). This certainly deserves a work-up by your doctor. An x-ray and blood work will go a long way to ruling out an infection. Breathing tests can help rule out asthma. A good history, physical, and medication review will also be required to get to the bottom of this.