We have implemented a systematic LSDA and $\mathrm{LSDA}+U$ study of the double perovskites $A_2{\mathrm{FeReO}}_6$ $(A=\mathrm{B}\mathrm{a},\mathrm{S}\mathrm{r},\mathrm{C}\mathrm{a})$ and Sr${}_{2}M$MoO${}_6$ $(M=\mathrm{C}\mathrm{r},\mathrm{M}\mathrm{n},\mathrm{F}\mathrm{e},\mathrm{C}\mathrm{o})$ for understanding of their intriguing electronic and magnetic properties. The results suggest a ferrimagnetic (FiM) and half-metallic (HM) state of $A_2{\mathrm{FeReO}}_6$ $(A=\mathrm{B}\mathrm{a},\mathrm{S}\mathrm{r})$ due to a $\mathrm{pdd}-\pi$ coupling between the down-spin ${\mathrm{Re}}^{5+}/{\mathrm{Fe}}^{3+}$ $t_{2g}$ orbitals via the intermediate $\mathrm{O}{2p}_{\pi }$ ones, also a very similar FiM and HM state of ${\mathrm{Sr}}_2{\mathrm{FeMoO}}_6.$ In contrast, a decreasing Fe $t_{2g}$ component at Fermi level ${(E}_F)$ in the distorted ${\mathrm{Ca}}_2{\mathrm{FeReO}}_6$ partly accounts for its nonmetallic behavior, while a finite $\mathrm{pdd}-\sigma$ coupling between the down-spin ${\mathrm{Re}}^{5+}/{\mathrm{Fe}}^{3+}{e}_g$ orbitals being present at $E_F$ serves to stabilize its FiM state. For ${\mathrm{Sr}}_2{\mathrm{CrMoO}}_6$ compared with ${\mathrm{Sr}}_2{\mathrm{FeMoO}}_6,$ the coupling between the down-spin ${\mathrm{Mo}}^{5+}/{\mathrm{Cr}}^{3+}$ $t_{2g}$ orbitals decreases as a noticeable shift up of the ${\mathrm{Cr}}^{3+}3d$ levels, which is likely responsible for the decreasing $T_{\mathrm{C}}$ value and weak conductivity. Moreover, the calculated level distributions indicate a ${\mathrm{Mn}}^{2+}\left({\mathrm{Co}}^{2+}\right)/{\mathrm{Mo}}^{6+}$ ionic state in ${\mathrm{Sr}}_2{\mathrm{MnMoO}}_6$ $\left({\mathrm{Sr}}_2{\mathrm{CoMoO}}_6\right),$ in terms of which their antiferromagnetic insulating ground state can be interpreted. While orbital population analyses show that owing to strong intrinsic $\mathrm{pd}$ covalence effects, ${\mathrm{Sr}}_{2}M{\mathrm{MoO}}_6$ $(M=\mathrm{C}\mathrm{r},\mathrm{M}\mathrm{n},\mathrm{F}\mathrm{e},\mathrm{C}\mathrm{o})$ have nearly the same valence state combinations, as accounts for the similar M-independent spectral features observed in them.

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