This paper presents significant theoretical extensions and practical applications of the Unified Relativistic Theory of Gravity (URTG). We develop the mathematical framework to incorporate hyperbolic geometry into URTG, demonstrating the integral connection between motion trajectories and space geometry. The enhanced formulation includes modified field equations, an extended hyperbolic metric tensor, and refined gravitational lensing equations. We analyze URTG's predictions for gravitational lensing against historical data from the 1919 Eddington eclipse expedition, showing alignment with observational evidence while suggesting subtle deviations that could be detected in future high-precision measurements. The paper addresses the treatment of singularities in URTG, demonstrating how the theory reinterprets black holes, the Big Bang, as well as Wormholes, as transitions between frame-dependent and frame-independent states, potentially resolving the black hole information paradox. Additionally, we explore gravitational waves through URTG's causal structure framework, deriving wave solutions from the Causality Emergence Equation that maintain consistency with current observations while offering new insights into gravity's quantum nature. These theoretical developments extend URTG's explanatory power while maintaining its foundational principles of relational space-time and emergent phenomena. The work provides testable predictions that could differentiate URTG from standard general relativity in future experiments, particularly in scenarios involving strong gravitational fields or quantum-scale effects.