For graphene/copper (Gr/Cu) composites, achieving high-quality interfaces between Gr and Cu (strong interfacial bonding strength and excellent electron transport performance) is crucial for enabling their widespread applications in electronic devices. This study employs first-principles calculations and the nonequilibrium Green's function method to systematically investigate the mechanical and electrical conductivity properties of Cu(111)/Gr/Cu(111) interfaces with various stacking sequences and different forms of Gr. For these interface systems, the binding energy, separation work, charge transfer, and electrical conductivity across the interface were obtained. The results show that the top-fcc interface exhibits superior interfacial properties, characterized by relatively high binding energy (-3.00 eV/C atom) and separation work (≥0.78 J/m2), a small interfacial distance (2.85 Å), and enhanced electron transport capacity (2.12 G0/nm2). A bilayer form of Gr significantly reduces electronic conductance across the Gr/Cu interface by nearly 2.46 orders of magnitude. Furthermore, point defects in Gr, especially single-vacancy defects, disrupt the traditional trade-offs between mechanical and electrical performance, simultaneously enhancing mechanical performance by 7.50-124.36% and electrical performance by 33.02%. Additionally, stress mechanisms have been proposed to further enhance the interfacial electrical conductivity of Gr/Cu composites. The present study provides a theoretical basis for exploring the engineering applications of Gr/Cu composite materials.