For all sums of eigenfunctions of a semiclassical Schrödinger operator below some given energy level, this paper proves that the ratio of the L² norm on R^d over the L² norm on any given open set is bounded by exp(C/h) for some positive C in the semiclassical limit h tends to 0. Corresponding estimates on a compact manifold are also given. They generalize the unique continuation estimate of Lebeau, with Jerison, Robbiano and Zuazua, on sums of classical eigenfunctions of the Laplacian on a compact manifold below an eigenvalue threshold as this threshold tends to infinity. The main tools are semiclassical Carleman estimates following Lebeau, Robbiano and Burq. For sums of classical Hermite functions, or for sums of classical eigenfunctions of homogeneous polynomial potential wells, a similar unique continuation estimate from a cone is deduced. It applies to the null-controllability from a cone of the heat semigroups corresponding to these Schrödinger operators, with a sharp cost estimate of fast control, following a new version of the strategy of Lebeau and Robbiano.