Sharp, burning pain left flank side near rib cage, can't take deep breaths, also burns towards my lower back...! any idea's?

Neurocostal . Sharp ; burning pain seems neuropathic and radiation pattern around flank of lower chest suggests neural impingement at costal vertebral junction. Inspiration ; costal lift would be expected to intensify this pain. Perhaps a rib-vertebra subluxation is at play. Absence of cough, fever, ; shortness of breath discount pneumothorax ; infection. Pleurisy certainly a concern. Doctor's exam a must.
Could be's. It could be pleurisy or inflammation of the lining of the lung and chest, could be the ribs and muscles, could be the spine and the nerves being irritated, could be the kidneys especially if infected, could be the pancreas. In short, there are too many could be's. It is better to see a doctor, be examined and proper tests can be requested to find out what's wrong.