Alice-Bob systems offer a powerful framework for describing nonlocal correlations between spatially separated physical events. In this paper, we investigate nonlinear wave structures of the second-order Alice-Bob Benjamin-Ono system, an integrable nonlocal model for two-place correlated phenomena. Using the Kadomtsev-Petviashvili hierarchy reduction method, we derive explicit Gram-determinant breather solutions and construct higher-order breather and breather-soliton interactions. By combining block-determinant formulations with long-wave limits, we obtain semi-rational solutions that describe interactions among breathers, solitons, and lump waves. In addition, through bilinear theory and Schur polynomial techniques, we derive general higher-order rogue wave solutions in compact determinant form. These solutions exhibit rich geometric patterns, including triangular and polygonal rogue wave configurations, whose orientations are controlled by free complex parameters while preserving peak amplitudes. The results reveal diverse interaction dynamics and complex localized wave structures in the second-order system. This unified determinant framework significantly enriches the family of exact solutions and provides new insights into nonlinear wave dynamics in nonlocal integrable systems.