Recent advancements in 3D printing technologies have made it possible to fabricate intricate lattice architectures with high precision. These lattices can now be utilized to design lightweight sandwich structures that serve multiple functions. To enhance the impact loading performance of these structures, it is crucial to understand how the lattice's topological properties, particularly those with minimal surface attributes like periodic or stochastic Primitive and Gyroid triply periodic minimal surfaces (TPMS) and spinodal-like stochastic cellular materials, associate with the mechanical properties of sandwich structures while keeping the skin thickness fixed. Thus, this paper explores the low-velocity impact behavior of various sheet/shell-based minimal surface-latticed cores of sandwich structures with woven composite skins. The elasto-plastic-damage numerical simulations consider lattice core periodicity, randomness, and anisotropy while keeping the relative density constant. Core lattice randomness and anisotropy are designed using the Gaussian Random Field (GRF) method for spinodal-based stochastic cellular materials and stochastic TPMS. The simulation results showed that the periodic Primitive-lattice core exhibits high out-of-plane shearing strength, enabling the sandwich structure to demonstrate the highest perforation limit. GRF spinodal-based core achieved the highest peak load due to its anisotropic mechanical properties. However, the post-yielding bending of the lattice sheet limited its ability to resist perforation, and absorb and dissipated energy. Interestingly, the stochastic Gyroid TPMS topology, with its inherent densely-distributed microstructure, showed high sensitivity to loading rate, resulting in enhanced energy absorption and dissipation of the sandwich structure. These findings offer valuable insights for optimizing multifunctional sandwich structures with superior impact performance and their design for additive manufacturing.