We study the entropy and Fisher information of mixtures of centered Gaussian random variables (with respect to the variance). First, we prove that if $X_1, X_2$ are independent scalar Gaussian mixtures, then the entropy of $\sqrt{t}X_1 + \sqrt{1-t}X_2$ is concave in $t \in [0,1]$, thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this statement, which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). Secondly, we establish rates of convergence for the Fisher information matrix of the sum of weighted i.i.d.~Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions. These are obtained by showing that the Fisher information matrix is operator convex as a matrix-valued function acting on densities of mixtures in $\mathbb{R}^d$, extending a result of Bobkov (2022).