when you say emf, it doesnt give energy to charges, but it pumps charges to the ends to make the potential different, when both ends are connected the charges flow from one end to other.
the basic about an inductor is it doesnt allow a suden change in its current, even you appply a suden voltage accross. same way when you try to sudenly break its current(by switch etc) it tries to keep it constant thus produces high voltage.
the wave you have posted is a steady AC that goes through a inductance, the voltage and curent waves. when you (the source) try to increase the voltage teh curent doesn't follow imediately, it takes time to increse because of the megnetic action as you said, same way when the voltage decreases it doesnt follow imediate response, it gives some lagging.
if you talk about charge flow you shuld term it as current (rate of flow of charge).