The performance of modern lithium-sulfur (Li/S) battery systems critically depends on the electrolyte and solvent compositions. For fundamental molecular insights and rational guidance of experimental developments, efficient and sufficiently accurate molecular simulations are thus in urgent need. Here, we construct a molecular dynamics (MD) computer simulation model of representative state-of-the art electrolyte-solvent systems for Li/S batteries constituted by lithium-bis(trifluoromethane)sulfonimide (LiTFSI) and LiNO3 electrolytes in mixtures of the organic solvents 1,2-dimethoxyethane (DME) and 1,3-dioxolane (DOL). We benchmark and verify our simulations by comparing structural and dynamic features with various available experimental reference systems and demonstrate their applicability for a wide range of electrolyte-solvent compositions. For the state-of-the-art battery solvent, we finally calculate and discuss the detailed composition of the first lithium solvation shell, the temperature dependence of lithium diffusion, as well as the electrolyte conductivities and lithium transference numbers. Our model will serve as a basis for efficient future predictions of electrolyte structure and transport in complex electrode confinements for the optimization of modern Li/S batteries (and related devices).