The double perovskites ${\mathrm{La}}_{2}M{\mathrm{RuO}}_{6+\ensuremath{\delta}}$ contain ${M}^{2+}$ ions for $M=\mathrm{Mg},$ Zn, Co, and Ni and for $M=\mathrm{Mn}$ and Fe, ${M}^{3+}$ ions. Annealed samples have ordered ${M}^{2+}$ and Ru(IV), but samples with ${M}^{3+}$ ions are atomically disordered and remain oxidized $(\ensuremath{\delta}g0)$ after annealing in a ${\mathrm{N}}_{2}$ atmosphere. A comparison of quenched, air-annealed, and ${\mathrm{N}}_{2}$-annealed ${\mathrm{La}}_{2}{\mathrm{MgRuO}}_{6+\ensuremath{\delta}}$ samples showed a persistence of a few cation vacancies in the presence of oxygen vacancies $(\ensuremath{\delta}l0).$ In quenched samples, oxygen vacancies are preferentially located between two Ru atoms, where they form a deep two-electron trap state, whereas the two-electron trap state formed at an oxygen vacancy between an ${M}^{2+}$ and Ru(IV) is shallow. Magnetic as well as transport data indicate the \ensuremath{\pi}-bonding $4d$ electrons at the low-spin Ru atoms occupy itinerant-electron states of a ${\ensuremath{\pi}}^{*}$ band even in the atomically ordered samples, but strong correlations introduce magnetic transitions among the ${\ensuremath{\pi}}^{*}$ electrons. The $M=\mathrm{Co}$ and Ni samples exhibit a magnetic transition at some, if not all, of the ${\ensuremath{\pi}}^{*}$ electrons on the Ru array below a ${T}_{\mathrm{irr}}$ independent of magnetic ordering on the ${M}^{2+}$ ions, and below a ${T}_{N}\ensuremath{\approx}26\mathrm{K}l{T}_{\mathrm{irr}},$ antiferromagnetic ordering of the ${M}^{2+}$-ion spins suppresses any spin on the intervening Ru(IV). The ${M}^{2+}{:e}^{2}\ensuremath{-}\mathrm{O}\ensuremath{-}\mathrm{Ru}(\mathrm{IV})\ensuremath{-}\mathrm{O}{\ensuremath{-}M}^{2+}{:e}^{2}$ antiferromagnetic superexchange interaction is stronger than the ferromagnetic ${M}^{2+}{:e}^{2}\ensuremath{-}\mathrm{O}\ensuremath{-}\mathrm{Ru}(\mathrm{IV}{):e}^{0}$ interaction because of a weak intraatomic exchange with the ${\ensuremath{\pi}}^{*}$-electron spins on the Ru(IV) atoms. On the other hand, disordered ${\mathrm{La}}_{2}{\mathrm{MnRuO}}_{6.17(1)}$ is ferromagnetic with a magnetization at 5 K of $2.85{\ensuremath{\mu}}_{B}$ per formula unit (f.u.) in a magnetic field of 50 kOe. This finding is interpreted with a model in which the \ensuremath{\pi}-bonding orbitals on both the Mn and Ru are coupled to form a common ferromagnetic ${\ensuremath{\pi}}^{*}$ band in which only the antibonding electrons are not spin paired. The strong next-nearest-neighbor interaction between Ru atoms made manifest in the ordered double perovskites provides an explanation of why the ${\ensuremath{\pi}}^{*}$ bandwidth of the perovskite system ${\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{RuO}}_{3}$ may increase with $x.$