Since the first report of graphdiyne nanomembranes synthesis in 2010, different novel graphdiyne nanosheets have been fabricated. In a latest experimental advance, triphenylene-graphdiyne (TpG), a novel two-dimensional (2D) material was fabricated using the liquid/liquid interfacial method. In this study, we employed extensive first-principles simulations to investigate the mechanical/failure, thermal stability, electronic and optical properties of single-layer TpG. In addition, we predicted and explored the properties of nitrogenated-, phosphorated- and arsenicated-TpG monolayers. Our results reveal that TpG, N-TpG, P-TpG and As-TpG nanosheets can exhibit outstanding thermal stability. These nanomembranes moreover were found to yield linear elasticity with considerable tensile strengths. Notably, it was predicted that monolayer TpG, As-TpG, P-TpG and N-TpG show semiconducting electronic characters with direct band-gaps of 1.94 eV, 0.88 eV, 1.54 eV and 1.91 eV, respectively, along with highly attractive optical properties. We particularly analyzed the application prospect of these novel 2D materials as anodes for Li-ion batteries. Remarkably, P-TpG and N-TpG nanosheets were predicted to yield ultrahigh charge capacities of 1979 mAh/g and 2664 mAh/g, respectively, for Li-ions storage. The acquired results by this work suggest TpG based nanomembranes as highly promising candidates for the design of flexible nanoelectronics and energy storage devices.