DragonSF Bela w/o Pepper and external Power:
Bela w/o Pepper and USB power:
Power supply noise (regardless of whether it's USB power or barrel jack/Eurorack) will leak into the audio, after being largely attenuated by the power management IC's buck converter and the audio codec's 44dB PSRR, so these are not too surprising if the USB power supply is bad enough.
In all these measurements, it makes very little sense to look at the audio output of Pepper when there is no audio being generated: in that case the codec's outputs - connected to the NE5532's inputs - are powered down and so anything you measure is meaningless. If you want to measure the noise floor, have the audio running and send out zeros instead, like you do in the last measurements.
DragonSF external power and signal (no idea, why there is some noise on the left channel, but I can't hear it - possible signal distortion)
This looks like harmonic distortion perhaps? It's puzzling that it only affects one channel, though, as the two are meant to be symmetrical, but also I don't see how any error in the component values you may have done during assembly of the PCB would result in this sort of behaviour. Perhaps yet another bad NE5532?
DragonSF The only solution (unless someone finds a remedy on the Bela board) is, to use external power.
I'd expect while the board is in the rack that it is always powered by the rack. If you have both USB and rack power on then the latter will always take priority.