Higher-order lattice Boltzmann (LB) pseudopotential models have great potential for solving complex fluid dynamics in various areas of modern science. The discreteness of the lattice discretization makes these models an attractive choice due to their flexibility, capacity to capture hydrodynamic details, and inherent adaptability to parallel computations. Despite those advantages, the discreteness makes high-order LB models difficult to apply due to the larger lattice structure, for which basic fundamental properties, namely diffusion coefficient and contact angle, remain unknown. This work addresses this by providing general continuum solutions for those two basic properties and demonstrating these solutions to compare favorably against known theory. Various high-order LB models are shown to reproduce the sinusoidal decay of a binary miscible mixture accurately and consistently. Furthermore, these models are shown to reproduce neutral, hydrophobic, and hydrophilic contact angles. Discrete differences are shown to exist, which are captured at the discrete level and confirmed through droplet shape analysis. This work provides practical tools that allow for high-order LB pseudopotential models to be used to simulate multicomponent flows.