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@MarcusMüller: not true; the observed chip/symbol rates will also be distorted by the same proportion as the carrier frequency. The receiver observes the signal as if it were resampled (at its original RF frequency) by a factor of $\frac{1]{1+\frac{v}{c}}$, where $v$ is the radial component of the velocity from transmitter to receiver. So, if the transmitter is traveling toward the receiver, each chip will appear to be a little shorter.
There can be pitfalls with this approach in the presence of noise. Squaring a signal will reduce the signal-to-noise ratio, so it can negatively impact your detection performance when the input SNR is sufficiently low. I'm not sure what you mean by "Signal is 100% DC" though. To answer your final question, no, you aren't gaining any useful resolution, because there is no additional information in the squared signal that wasn't there before. I'm not clear on exactly what your end goal is, so I'm not sure of a different approach to suggest.
And so it follows that the lowpass filter, which is designed to have approximately unity gain at DC, has coefficients that sum to 1, while the highpass filter, which has high attenuation at DC, has coefficients that sum to nearly zero.
It's just an indication that the dataset has a lowpass characteristic; that is, as frequency increases, the spectral content decays toward zero. This is a common attribute of signals, especially signals that have had anti-aliasing filtering applied before digitization. It doesn't have anything to do with Welch's method; for example, try generating white Gaussian noise and see that the PSD estimate is nearly flat.
Another way to look at it is that (10) is the same result that you get for standard BPSK (i.e. antipodal signaling), taking spread spectrum out of the equation. Direct-sequence spread spectrum or chirp spread spectrum do not affect the bit error rate. Indeed they cannot affect the BER of the optimal receiver, due to the principle of reversibility (ref: Principles of Communication Engineering by Wozencraft and Jacobs). Application of the spreading waveform (sequence or chirp) is reversible, so an optimal receiver can just undo the operation first and obtain the same performance.
It's not clear to me where the "sudden change" that you're looking to identify appears in the plot. I'm not familiar with the details of your end application.
