The de Bruijn identity states that Fisher information is equal to a half of the time-derivative of Shannon differential entropy along heat flow. In the same spirit, a generalized version of Fisher information, which we term the Rényi--Fisher information, is defined as a half of the time-derivative of Rényi differential entropy along heat flow. Based on this Rényi--Fisher information, we establish several sharp Rényi-entropic isoperimetric inequalities, which generalize the classic entropic isoperimetric inequality to the Rényi setting. Utilizing these isoperimetric inequalities, we extend the classical Cramér--Rao inequality from Fisher information to Rényi--Fisher information. We then use these generalized Cramér--Rao inequalities to determine the signs of derivatives of Rényi entropy along heat flow, strengthening existing results on the complete monotonicity of Rényi entropy. We lastly explore the implications of our Rényi-entropic isoperimetric inequalities for entropy power inequalities. We demonstrate that, unlike in the Shannon entropy case, the classic entropy power inequality does not admit a direct extension to Rényi entropy without introducing additional exponents or scaling factors. Furthermore, we establish a sharp Rényi entropy power inequality involving a scaling factor under the assumption that one of two independent random vectors is Gaussian.