I am eternally confused even after repeated learning about the mutual independence between affine connections and the metric tensor and parallel transport. Given any one of them, can I recover the rest?? I know that given any smooth manifold, there are infinitely many affine connections defined. What exactly are the requirements for the existence of each of the three mentioned above?? Wikipedia says that connections and parallel transport can each help define the other. I know this is a fairly elementary question, but I am looking for an answer which will help me recall it again without confusion. 

Lastly, is the Lie bracket a necessity or sufficient for defining any of these? Hope I haven't crammed too many questions into one.