We show that the Bergman metric on ball quotients $\mathbb{B}^2/\Gamma$ is Kähler-Einstein if and only if $\Gamma$ is trivial, leading to a characterization of the unit ball among certain two-dimensional Stein spaces, confirming a version of Cheng’s conjecture. We also relate the boundary type of two-dimensional Stein spaces to the local algebraic degree of their Bergman kernel, characterizing ball quotients via the local rationality of the Bergman kernel. Finally, we derive the rotational symmetry properties of certain domains in $\mathbb{C}^n$ from the orthogonality of holomorphic monomials in their Bergman spaces.