Simulations of nano-to micro-meter scale fluidic systems under thermal gradients require consistent mesoscopic methods accounting for both hydrodynamic interactions and proper transport of energy. One such method is dissipative particle dynamics with energy conservation (DPDE), which has been used for various fluid systems with non-uniform temperature distributions. Despite the success of the method, existing integration algorithms have shown to result in an undesired energy drift, putting into question whether the DPDE method properly captures properties of real fluids. We propose a modification of the velocity-Verlet algorithm with local energy conservation for each DPDE particle, such that the total energy is conserved up to machine precision. Furthermore, transport properties of a DPDE fluid are analyzed in detail. In particular, an analytical approximation for the thermal conductivity coefficient is derived, which allows the selection of a specific value a priori. Finally, we provide approximate expressions for the dimensionless Prandtl and Schmidt numbers, which characterize fluid transport properties and can be adjusted independently by a proper selection of model parameters, and therefore, made comparable with those of real fluids. In conclusion, our results strengthen the DPDE method as a very robust approach for the investigation of mesoscopic systems with temperature inhomogeneities. arXiv:1907.06186v1 [cond-mat.soft]