Chronic cough can be caused by many factors from either the upper airway, such as the sinuses, or the lower airways, such as the lungs. Since the chest x-ray is normal, perhaps a lung function test and a CT scan of the sinuses might provide some insight as to the cause of this problem. I would continue to work with your daughter
Increased vascular markings on a chest xray usually indicates enlarged arteries of the heart. A heavy shadow indicates a possible problem with the pulmonary artery, or the artery that moves blood to or from the lungs. Without actually viewing the xray, these findings can be anything from bad positioning by the tech that took the xrays to an enlarged heart indicating congestive heart failure, a heart defect or possibly left sided heart failure. More tests are indicated to positively identify the problem.
has your dtr seen a pediatric pulmonologis? also have they tried culturing the mucous? this seems an awfully long time to have green mucous. have they done a cat scan of her sinuses or worked her up for allergies? about the chest xray i really don't know what it means, but i would see a specialist good luck