Our unified superfast algorithms for solving Toeplitz, Hankel, Vandermonde, Cauchy, and other structured linear systems of equations with integer coefficients combine Hensel's symbolic lifting and numerical iterative refinement and run in nearly optimal randomized Boolean time for both solution and its correctness verification. The algorithms and nearly optimal time bounds are extended to some fundamental computations with univariate polynomials that have integer coefficients. Furthermore, we develop lifting modulo powers of two to implement our algorithms in the binary mode within a fixed precision.