Validity of superposition when summing powers from each harmonic
Why the total average power equals the sum of the powers from each component if the superposition theorem can't be applied to power?

This is a consequence of Parseval's theorem, which says that the the integral of the square of a function is equal to the sum of the square of its Fourier series.
Put another way, the power in the signal is the same whether you represent it in the time domain or the frequency domain.