Graphdiyne and graphyne are carbon-based two-dimensional (2D) porous atomic lattices with outstanding physics and excellent application prospects for advanced technologies, like nanoelectronics and energy storage systems. During the last few years, B- and N-graphdiyne nanomembranes were experimentally realized. Motivated by the latest experimental advances, in this work, we predicted novel N-, B-, P-, Al-, As-, and Ga-graphdiyne/graphyne 2D lattices. We then conducted density functional theory simulations to obtain the energy minimized structures and explore the mechanical, thermal stability, electronic and optical characteristics of these novel porous nanosheets. The acquired theoretical results reveal that the predicted carbon-based lattices are thermally stable. It was moreover found that these novel 2D nanostructures can exhibit remarkably high tensile strengths or stretchability. The electronic structure analysis reveals a semiconducting electronic character for the predicted monolayers. Moreover, the optical results indicate that the first absorption peaks of the imaginary part of the dielectric function for these novel porous lattices along the in-plane directions are in the visible, IR and near-IR (NIR) ranges of light. This work highlights the outstanding properties of graphdiyne/graphyne lattices and recommends them as promising candidates in the design of stretchable energy storage and nanoelectronics systems.