No one is pretending fermions are classical, I'm remaining in the domain of classical quantum mechanics, there is no relativity, there is no spin and therefore no fermion or boson statistics. The doppler effect is a direct consequence of the wave nature of the functions that describe both light and particles (as probability waves). Plus energy and wave frequency have always been related. What quantum mechanics adds is that particles have discrete energy levels. Looking at the photoelectric effect in particular you can prove that energy is directly proportional to the frequency of light. I don't understand what wasn't clear on my explanation, in case reply to me and ask please so that I can better explain

For the case of alpha and beta radiation (gamma and neutron too actually), yes, it would be red/blue-shifted. You'd see that in energy shifts since E~hc/λ. You'd still be able to discern the type of radiation via the usual means (charge, penetration depth, interaction,...) but its energy spectrum would be shifted

Yes, radiation in relative motion towards or from you is "pitch shifted". The concept of redshift comes from this, if you do the calculations you have that if a light source moving away from you is "redshifted", i.e. the wavelengths are shifted towards the red (they get longer) with respect to a still source. The opposite is called "blueshift", since when the source moves towards you it's shifted towards the blue (shorter wavelengths). In experimental physics the blueshift/redshift is widely used to measure astronomical distances (you know, that redshift thingy that astrophysicists use to indicate cosmological distances, that's expansion-induced redshift), and also it's used to evaluate relative speeds. As an example, if you look at the spectrum of stars in Andromeda, since you more or less basically know already the theoretical (still) values of the spectrums, what you get is that differences from the theoretical value are exactly what you'd expect in a rotating disk, with one side coming towards you (blueshift) and one going away from you (redshift). With appropriate measurements of this you can therefore know how fast it's rotating and how quick is coming towards or further from you. Note that this shift is always there when there's relative movement, therefore the cutoff velocity for seeing it depends only on instrument sensibility, don't expect hopping on a very fast plane and starting to see the world in front of you bluer and the one behind you redder 😅