$3d-5d$ transition-metal double perovskites exhibit diverse magnetic and electronic phases arising from the interplay of electron correlation ($U$) and spin-orbit coupling (SOC). Using first-principles calculations, we investigate the contrasting magnetic and charge states of ${\mathrm{Ba}}_{2}X{\mathrm{IrO}}_6(X=\mathrm{Ni},\mathrm{Co},\mathrm{Fe})$, incorporating the combined effects of $U$ and SOC. Structural and magnetic ground-state searches show that all compounds adopt a rocksalt structure, with ferromagnetic ordering in ${\mathrm{Ba}}_2{\mathrm{FeIrO}}_6$ and distinct antiferromagnetic phases (AFM-1 and AFM-2) in ${\mathrm{Ba}}_2{\mathrm{NiIrO}}_6$ and ${\mathrm{Ba}}_2{\mathrm{CoIrO}}_6$, respectively. Dynamical mean-field theory (DMFT) captures dynamic correlations and confirms nominal Ir charge states of $+6(X=\mathrm{Ni},\mathrm{Co})$ and $+5(X=\mathrm{Fe})$, leading to half-metallic (Fe), semiconducting (Ni), and metallic (Co) behavior. DMFT spectral functions corroborate the transport characteristics, while impurity charge analysis clarifies Ir charge variation. Exchange interactions within a classical Heisenberg model reveal ferromagnetic nearest-neighbor ($J_{X-\mathrm{Ir}}$) and antiferromagnetic next-nearest-neighbor ($J_{X-X}, J_{\mathrm{Ir}-\mathrm{Ir}}{J}_{\mathrm{Ir}-\mathrm{Ir}}$) couplings, with the latter stabilizing AFM order in the Ni and Co compounds. Benchmarking against available experimental data for ${\mathrm{Ba}}_2{\mathrm{NiIrO}}_6$ provides a comprehensive framework for exploring related double perovskites and guide future experimental studies.

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• Charge order
• Spin-orbit coupling
• Perovskites
• Strongly correlated systems
• Ab initio calculations
• DFT+DMFT