Understanding alternating current (sine wave)
Since you prefer an Alternating current analogy , I will try to explain both of your concerns with the help of a simple parallel LC circuit.







simulate this circuit &ndash; Schematic created using CircuitLab

Suppose initially that capacitor was charged to some charge Q. In this configuration, it will begin to discharge through the inductor. Thus it will start to lose energy which is stored in the form of Electric field.

U =  ( Q^2 ) / 2C


Now flowing charges constitute current. Current will increase in the circuit. As a result the energy stored in the inductor, which is in the form of Magnetic field will start to increase.

U = ( L * I^2) / 2


Ultimately, the capacitor will lose all of its charge and energy will be completely stored in the form of magnetic field. 

But wait, inductor has one special property : It prevents any abrupt change. So what does that mean?

Although energy is completely magnetic now, the flow of current cannot stop, because inductor doesnt allow it to suddenly stop. 

So now the capacitor comes into play. It is discharged, so charge begins to pile up on its plates due to current flowing in the circuit. And capacitor begins to charge itself, and the cycle continues.

Now regarding your second query, why is the rate different. If you observe carefully, that should be pretty much expected. Lets take capacitor for example.

Initially (after the capacitor has discharged) , it has no or very little charge. So piling up more charge on its plates doesnt really require much effort. As charge begins to pile up, the electrons already present on the plate begin to oppose the incoming charge. So it basically requires more time to put charges on the plate. So you see the graph initially jumps pretty quickly, but then the rate drops as more charges are put on the plates.