Extracting operationally meaningful quantities such as von Neumann entropy and mutual information is central to characterizing many-body quantum systems, yet experiments and numerics often provide direct access only to integer Rényi entropies. Whereas conventional extrapolation relies on a prescribed fitting ansatz, in PRL 137, 100202 we introduced an alternative approach based on stabilized analytic continuation (SAC), which avoids such an ansatz and is inherently robust to noise. Here, we further develop this framework in the noiseless setting and benchmark its performance across several nontrivial many-body problems. We first show that, besides the intrinsic ambiguity of reconstructing the von Neumann entropy from finitely many Rényi samples, analytic continuation can also fail because of genuine nonanalyticities in the Rényi function arising from zeros of Trρᶻ. We derive universal zero-free domains for Trρᶻ and systematically sharpen these bounds using additional spectral information about ρ. We then benchmark the performance of SAC in three settings: (i) extracting topological entanglement entropy from DMRG Rényi data for the toric code and Kagome Heisenberg models; (ii) reconstructing the mutual information between disjoint intervals in a (1+1)d compact-boson CFT; and (iii) recovering finite-time ballistic growth of the von Neumann entropy from integer Rényi entropies that cross over toward subballistic growth in diffusive quantum dynamics. In the latter setting, analytic continuation is expected to fail in the long time limit. We illustrate this mechanism with a physically motivated two-sector model, where the asymptotic separation between von Neumann and higher-Rényi growth is accompanied by a zero of Trρᶻ pinching the real axis at z=1, thereby producing a first-order Rényi phase transition.