Sufficiency, ancillarity and the Cramer–Rao bound — reading Basu's theorem carefully

The following are some statements: (I) If T is a boundedly complete sufficient statistic for $\theta$ and S is an ancillary statistic for $\theta$, then T and S are independent. (II) An ancillary statistic may or may not be independent of a sufficient statistic. (III) It is possible to have variance smaller than that of the Cramer-Rao lower bound at some points of the parameter space. (IV) The parametric function need not be differentiable for the Cramer-Rao lower bound to hold.