Context. The Euclid mission of the European Space Agency will deliver weak gravitational lensing and galaxy clustering surveys that can be used to constrain the standard cosmological model and extensions thereof. Aims. We present forecasts from the combination of the Euclid photometric galaxy surveys (weak lensing, galaxy clustering, and their cross-correlations) and its spectroscopic redshift survey with respect to their sensitivity to cosmological parameters. We include the summed neutrino mass, Σmν, and the effective number of relativistic species, Neff, in the standard Λ CDM scenario and in the dynamical dark energy (w0waCDM) scenario. Methods. We compared the accuracy of different algorithms predicting the non-linear matter power spectrum for such models. We then validated several pipelines for Fisher matrix and Markov chain Monte Carlo (MCMC) forecasts, using different theory codes, algorithms for numerical derivatives, and assumptions on the non-linear cut-off scale. Results. The Euclid primary probes alone will reach a sensitivity of σ (Σmν = 60 meV) = 56 meV in the Λ CDM+Σmν model, whereas the combination with cosmic microwave background (CMB) data from Planck is expected to achieve σ (Σmν) = 23 meV, offering evidence of a non-zero neutrino mass to at least the 2.6 σ level. This could be pushed to a 4 σ detection if future CMB data from LiteBIRD and CMB Stage-IV were included. In combination with Planck, Euclid will also deliver tight constraints on Δ Neff < 0.144 (95%CL) in the Λ CDM+Σmν+Neff model or even Δ Neff < 0.063 when future CMB data are included. When floating the dark energy parameters, we find that the sensitivity to Neff remains stable, but for Σmν, it gets degraded by up to a factor of 2, at most. Conclusions. This work illustrates the complementarity among the Euclid spectroscopic and photometric surveys and among Euclid and CMB constraints. Euclid will offer great potential in measuring the neutrino mass and excluding well-motivated scenarios with additional relativistic particles.