The normal chest x-ray is reassuring. However you could have a little pleurisy along with the acute bronchitis. Pleurisy is inflammation or irritation of the pleura, the sac that holds the lungs. This can cause chest tightness and pain, which is felt more with deep breathing and coughing.
A urinary tract infection is the one problem that would explain both back and kidney pain along with burning when urinating. Your back pain could be pain referred from your urinary tract. You should be examined for a urinary tract infection and treated for it.