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. We present forecasts from the combination of these surveys on the sensitivity to cosmological parameters including the summed neutrino mass and the effective number of relativistic species in the standard CDM scenario and in a scenario with dynamical dark energy (CDM). We compare the accuracy of different algorithms predicting the nonlinear matter power spectrum for such models. We then validate several pipelines for Fisher matrix and MCMC forecasts, using different theory codes, algorithms for numerical derivatives, and assumptions concerning the non-linear cut-off scale. The Euclid primary probes alone will reach a sensitivity of 56meV in the CDM+ model, whereas the combination with CMB data from Planck is expected to achieve 23meV and raise the evidence for a non-zero neutrino mass to at least the level. This can be pushed to a detection if future CMB data from LiteBIRD and CMB Stage-IV are included. In combination with Planck, Euclid will also deliver tight constraints on (95%CL) in the CDM++ model, or when future CMB data are included. When floating , we find that the sensitivity to remains stable, while that to degrades at most by a factor 2. This work illustrates the complementarity between the Euclid spectroscopic and imaging/photometric surveys and between Euclid and CMB constraints. Euclid will have a great potential for measuring the neutrino mass and excluding well-motivated scenarios with additional relativistic particles.