The spin Nernst effect (SNE) and anomalous Nernst effect (ANE) convert a heat current generated by a temperature gradient to a spin current in nonmagnetic metals and ferromagnets, respectively. We present a diffusive spin transport theory of the combination of these thermoelectric effects in ferromagnet (F)/nonmagnet (N) heterostructures and their respective output signals. In a F/N bilayer, in the presence of an in-plane temperature gradient, electrical voltages are induced by the SNE in N and the ANE in the F layers, via the inverse spin Hall effect in the N. The analytical expression of the thermally driven spin Hall magnetoresistance (TSMR) output is derived, which captures both the local SNE contribution in the N and the nonlocal mixing contribution due to the ANE in the adjacent F. Interestingly, the SNE and ANE give additive contributions to the transverse TSMR voltages but subtract from each other for the longitudinal component. This anisotropic response suggests a possible means to extract the individual SNE and ANE contributions, as well as spin-dependent polarization parameters. In a F1/N/F2 trilayer, both SNE and ANE contribute to spin torques on the free F2 layer. We analyze the ferromagnetic resonance (FMR) of the free layer and show that both thermoelectric effects contribute additively to the FMR magnitude and linewidth for the considered magnetization configuration. Finally, we show that TSMR can provide a more sensitive experimental detection of SNE and ANE down to a threshold thermal gradient of the order of ${10}^{\ensuremath{-}6}$ K/nm.