Does an ideal capacitor dissipate power?
No, they don't dissipate energy, but they do store it.  So energy can flow into a capacitor, and remain there, and then flow back out.  Since power is the rate at which energy is used or moved, the power will be non-zero whenever energy is moving.  But it's not being dissipated (converted into heat).  It's just being moved around and stored.

It's important to remember the difference between instantaneous power and average power.  A 1 ohm resistor with 1 VRMS AC across it will dissipate an average power of 1 W, for instance, but the instantaneous power will vary with the waveform:



With a reactive element (capacitor or inductor) as the load, the power will fluctuate between positive and negative as energy flows in and out, but the average will be zero: