Should I combine the negative part of the spectrum with the positive one? When filtering sound I currently analyse only the positive part of the spectrum. From the mathematical point of view, will discarding the negative half of the spectrum impact significantly on my analysis?
Please consider only samples that I will actually encounter, not computer generate signals that are designed to thwart my analysis.
I know this question involves physics, biology and even music theory. But I guess the required understanding of mathematics is deeper than of those other fields of study.

Sound processing is achieved through Real signal samples. Therefore there is no difference in the phase and magnitude of the FFT, or DFT coefficients, from positive to negative part of the found spectrum.
So, to save us or the machine the burden of saving/analyzing twice the same information/data, one looks only to the positive side of the FFT/DFT. However, do take notice that when figuring out spectral energy, you must remember to multiply the density by two (accounting for the missing, yet equal, negative part).
