  Consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a
hyperbolic metric $\sigma$ of total area $A$. In this article, we study the
behavior of geometric and dynamical characteristics (e.g., diameter, Laplace
spectrum, Gaussian curvature and entropies) of nonpositively curved smooth
metrics with total area $A$ conformally equivalent to $\sigma$. For such
metrics, we show that the diameter is bounded above and the Laplace spectrum is
bounded below away from zero by constants which depend on $\sigma$. On the
other hand, we prove that the metric entropy of the geodesic flow with respect
to the Liouville measure is flexible. Consequently, we also provide the first
known example showing that the bottom of the $L^2$-spectrum of the Laplacian
cannot be bounded from above by a function of the metric entropy. We also
provide examples showing that our conditions are essential for the established
bounds.
