We present a detailed study of proton dynamics in the hydrogen-bonded superprotonic conductor CsHSO(4) from first-principles molecular dynamics simulations, isolating the subtle interplay between the dynamics of the O-H chemical bonds, the O center dot center dot center dot H hydrogen bonds, and the SO(4) tetrahedra in promoting proton diffusion. We find that the Grotthuss mechanism of proton transport is primarily responsible for the dynamics of the chemical bonds, whereas the reorganization of the hydrogen-bond network is dominated by rapid angular hops in concert with small reorientations of the SO(4) tetrahedra. Frequent proton jumping across the O-H center dot center dot center dot O complex is countered by a high rate of jump reversal, which we show is connected to the dynamics of the SO(4) tetrahedra, resulting in a diminished CsHSO(4)/CsDSO(4) isotope effect. We also find evidence of multiple time scales for SO(4) reorientation events, leading to distinct diffusion mechanisms along the different crystal lattice directions. Finally, we employ graph-theoretic techniques to characterize the topology of the hydrogen-bond network and demonstrate a clear relationship between the likelihood for diffusive jump events and the presence of connectivity configurations favoring linear network chains over closed network rings.