Graphdiyne nanomaterials are low density and highly porous carbon-based two-dimensional (2D) materials, with outstanding application prospects for electronic and energy storage/conversion systems. In two latest scientific advances, large-area pyrenyl graphdiyne (Pyr-GDY) and pyrazinoquinoxaline graphdiyne (PQ-GDY) nanosheets have been successfully fabricated. As the first theoretical study, herein we conduct first-principles simulations to explore the stability and electronic, optical and mechanical properties of Pyr-GDY, N-Pyr-GDY, PQ-GDY and N-Pyr-GYN monolayers. We particularly examine the intrinsic properties of PQ-graphyne (PQ-GYN) and Pyr-graphyne (Pyr-GYN) monolayers. Acquired results confirm desirable dynamical and thermal stability and high mechanical strength of these novel nanosheets, owing to their strong covalent networks. We show that Pyr-based lattices can show high stretchability. Analysis of optical results also confirm the suitability of Pyr- and PQ-GDY/GYN nanosheets to adsorb in the near-IR, visible, and UV range of light. Notably, PQ-GDY is found to exhibit distorted Dirac cone and highly anisotropic fermi velocities. First-principles results reveal ultrahigh carrier mobilities along the considered nanoporous nanomembranes, particularly PQ-GYN monolayer is predicted to outperform phosphorene and MoS2. Acquired results introduce pyrenyl and pyrazinoquinoxaline graphyne/graphyne as promising candidates to design novel nanoelectronics and energy storage/conversion systems.