do power amps need regulated power supply
A power amplifier does not need a regulated supply depending on how it is specified.  Some may need a regulated supply to meet their stated power line noise output limits.  Others may need a regulated supply to deliver their maximum rated power without overheating.

It is certainly possible to design a audio power amplifier to a useful set of specifications while allowing its DC power voltage to fluctuate over some limits.  Allowing the power supply to ripple introduces two design challenges, to not let the ripple get into the output signal, and to make the amp work as specified over the full range of minimum to maximum power voltage.

Keeping the power supply noise out of the output signal is quantified as power supply rejection, usually expressed in dB.  Negative feedback is one important way to increase power supply rejection.  Take a look at the feedback equation and see how it inherently does this.  However, you don't want global feedback too high because you otherwise get into other problems, like transcient intermodulation (TIM) distortion.  The circuit topology is also important.  As a simplistic example, consider a emitter follower.  Changes in the collector voltage are greatly attenuated onto the emitter voltage output.

Making the amp work over a range of DC input voltage is mostly about being careful about dissipation.  Assuming the amp can only drive the output within its power supply range minus a little headroom, the lowest power voltage therefore dictates the maximum output power.  However, the most dissipation will occur at the highest power voltage.  You need to make sure the amp can handle the rated output power at the highest power voltage without overheating.  A class D amp deals with this issue nicely in that it is essentially a switching power supply that runs at audio rates.