Optimal Transport (OT) provides powerful geometric tools for comparing probability measures, but its practical application often hinges on efficient computation. This contribution highlights two distinct advancements in the field. First, we address the significant computational challenges in solving the regularized semi-discrete optimal transport (RSOT) problem, which involves mapping a continuous source measure to a discrete target measure. The bottleneck lies in evaluating the dual objective and gradient, requiring costly interactions between source integration points and numerous target points (N). We present a suite of numerical strategies within a finite element framework to drastically accelerate this process. Key techniques include distance-based interaction truncation facilitated by R-tree spatial indexing, computational caching to exploit temporal coherence, multilevel hierarchies for both source and target measures, and robust regularization parameter scheduling (epsilon-scaling). These methods substantially reduce the computational complexity, making large-scale RSOT feasible. Second, we demonstrate the application of OT principles, specifically discrete displacement interpolation, to enhance Reduced Order Models (ROMs) for complex nonlinear dynamical systems, particularly those dominated by advection and characterized by limited observational data. We propose a data augmentation strategy where synthetic, physically consistent snapshots are generated by interpolating between sparse checkpoints using pre-computed discrete OT plans. This enriched dataset improves the ROM's ability to capture nonlinear dynamics. The framework further incorporates a virtual-to-real time mapping for continuous-time predictions and employs a POD-based Gaussian Process Regression (POD-GPR) correction on the residuals to improve accuracy. While this ROM application utilizes discrete OT, the development of efficient solvers, such as those for RSOT, is crucial for broadening the applicability of OT across computational science. Together, these advancements showcase progress in both the fundamental computation and the practical application of OT methodologies.