We establish various analytic and geometric estimates for singular Kahler spaces based on the volume entropy bounds.  For any 3-dimensional projective variety X with klt singularities, we prove that every singular Kahler metric on X with bounded Nash entropy and Ricci curvature bounded below induces a compact RCD space homeomorphic to the projective variety X itself. Various compactness theorems are also obtained for 3-dimensional projective varieties with bounded Ricci curvature. Such results establish connections among algebraic, geometric and analytic structures of klt singularities in birational geometry and provide abundant examples of RCD spaces from algebraic geometry via complex Monge-Ampere equations.