We derive a closed-form joint distribution of the first arrival time (FAT) and first arrival position (FAP) in drift-diffusion molecular communication (MC) channels. In contrast to prior studies that analyze FAT or FAP in isolation, our framework explicitly captures the spatiotemporal coupling inherent in multidimensional transport. Building on this derivation, we compute the Fisher information matrix (FIM) and demonstrate that estimation accuracy for diffusivity scales proportionally with the spatial dimension, enabling increased sensitivity in higher-dimensional environments. Furthermore, we show that lateral drift -- which is unobservable from timing data alone -- can be recovered via a closed-form Maximum Likelihood Estimator (MLE) with a simple physical interpretation. Leveraging this spatial degree of freedom, we propose Drift Shift Keying (DSK), proving that joint receivers can reliably detect signals that are undetectable to timing-only receivers due to identical marginal FAT distributions. These results highlight the significant potential of spatiotemporal processing for future nanoscale communication and sensing.