We investigate generalized interacting dark matter-dark energy scenarios with a time-dependent coupling parameter, allowing also for freedom in the neutrino sector. The models are tested in the phantom and quintessence regimes, characterized by an equation of state $w_x<-1$ and $w_x>-1$, respectively. Our analyses show that for some of the scenarios the existing tensions on the Hubble constant $H_0$ and on the clustering parameter $S_8$ can be significantly alleviated. The relief is either due to textit{(a)} a dark energy component which lies within the phantom region; or textit{(b)} the presence of a dynamical coupling in quintessence scenarios. The inclusion of massive neutrinos into the interaction schemes does not affect neither the constraints on the cosmological parameters nor the bounds on the total number or relativistic degrees of freedom $N_{rm eff}$, which are found to be extremely robust and, in general, strongly consistent with the canonical prediction $N_{rm eff}=3.045$. The most stringent bound on the total neutrino mass $M_{ u}$ is $M_{ u}<0.116$ eV and it is obtained within a quintessence scenario in which the matter mass-energy density is only mildly affected by the presence of a dynamical dark sector coupling.
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