Volt-Sec-balance and Capacitor-Charge-balance
So the Voltage-average over one period is equal to Zero. It follows
  that the Current-average over one period is constant.


For an inductor \$V = L\dfrac{di}{dt}\$.

And a simple outcome of this is that there can be no change in current if the voltage is zero. 



Put differently, if the average voltage over a certain time period is zero then the current at the start of that period must be the same as the current at the end of that period.


  I have the same question for the CC-Balance for the behaviour of the
  Current and Voltage of the Capacitance


You need to unambiguously state the relationship you are trying to understand just like you did for the inductor (at the top of this answer).