question on diode in the Energy storing circuit
The answer to your question is more or less yes, but I'm not sure you're asking the right question.

1) Only when there's a voltage across and a current through the load is energy being delivered to the load, which you've drawn as a capacitor.

2) I_d is zero when the source voltage is too low.

3) The amount of energy delivered to the load depends on i_d and v(t)-v_d

The input signal does not 'have a power', as the power it delivers depends on the load, and the load the signal sees changes with the load's voltage, due to your diode.

The input signal has a voltage, and an output impedance, the latter complicating what happens to the voltage that appears across a load when a current flows.