In this paper, we introduce a family of two-dimensional group-V carbides with an ordered sequence of tetragons and hexagons (th-$\mathrm{C}{X}_{2}$, where $X=\text{P}$, As, and Sb). We demonstrate that th-$\mathrm{C}{X}_{2}$ monolayers exhibit robust energetic, dynamical, thermal, and mechanical stability. Our calculations show that the intrinsic structural anisotropy of the th-$\mathrm{C}{X}_{2}$ family induces strongly anisotropic mechanical, electronic, and optical behavior. These monolayers offer ultrahigh ultimate tensile strength, comparable with that of graphene, making them suitable for strain engineering of electronic and optical properties. They are semiconductors in nature, where th-${\mathrm{CP}}_{2}$, th-${\mathrm{CAs}}_{2}$, and th-${\mathrm{CSb}}_{2}$ possess quasidirect, direct, and indirect band gaps, respectively. The band gaps of th-${\mathrm{CP}}_{2}$ and th-${\mathrm{CAs}}_{2}$ are wide enough to provide the photogenerated energy required for the splitting of water. Besides, the positions of band edges are in alignment with the water oxidation and reduction potentials. For th-${\mathrm{CSb}}_{2}$, however, the suitable width of the band gap and the appropriate band edge positions for photocatalytic water splitting are achieved by strain engineering. Both indirect-to-direct and direct-to-indirect band gap transitions can be induced in th-$\mathrm{C}{X}_{2}$ compounds through strain engineering. The th-$\mathrm{C}{X}_{2}$ monolayers offer anisotropic high charge carrier mobility, which prolongs the average lifetime of charge carrier drift. They have good optical absorption ($\ensuremath{\sim}{10}^{5}$ ${\mathrm{cm}}^{\ensuremath{-}1}$) in the visible and ultraviolet regions of the light spectrum. $G{W}_{0}+$BSE (Bethe-Salpeter equation) calculations reveal that they exhibit strong excitonic effects where the first bright excitonic binding energy is calculated as 0.27, 0.52, and 0.22 eV for th-${\mathrm{CP}}_{2}$, th-${\mathrm{CAs}}_{2}$, and th-${\mathrm{CSb}}_{2}$, respectively. Having all these features in one package, the th-$\mathrm{C}{X}_{2}$ monolayers are among the best candidates for high-performance photocatalytic water splitting.