If your power conditioner has higher filtering designed for analog circuit and less for digital, then treat Class D amp as digital as far as the power conditioner is concerned. Class D design is by nature much noisier than pure Class A design. It took more than a decade for engineer to figure out how to reduce the noise and increase the switching frequency to the point where noise has significantly reduced and made irrelevant. If you recall, in the early days, Class D amps were publishing 0.1% THD+N and then people started to brag about 0.05% and then 0.01% was like a very big deal, since it was approaching Class A's THD+N.
But now who cares? Because even a low cost NuPrime amp can achieve 0.005% THD+N. By the way, you can't hear the difference as the THD+N goes below 0.05%. It has become so low that the instrument can't measure it. And in some design (Class A+D), we actually added back some "noise" as even order harmonics to make it sounded warmer.
Self oscillation is a naturally occurring phenomenon, see https://en.wikipedia.org/wiki/Self-oscillation
So if a circuit is arrange in certain way, a sine wave started to appear. In order to make the circuit oscillates, some external input (i.e. music signal) is required. Over the years, we have been improving this circuit to the point where it can oscillate at 700kHz. It is not easy to control this. Self oscillation circuit can become unstable (like spinning out of control) so it took many years to master this technique. And that's the reason only a handful of high end Class D designs existed in the world. The analogy would be street legal sports car. There are very few (street legal) cars that can run at top speed of 200m/h for example. To make the engine, drive train run at that kind of speed, the tire design, aerodynamic etc all have to meet the requirement. Likewise for amplifier circuit.