We present details of a continuous-time quantum Monte Carlo algorithm for the screened Hubbard-Fr\"ohlich bipolaron. We simulate the bipolaron in one dimension with arbitrary interaction range in the presence of Coulomb repulsion, computing the effective mass, binding energy, total number of phonons associated with the bipolaron, mass isotope exponent, and bipolaron radius in a comprehensive survey of the parameter space. We discuss the role of the range of the electron-phonon interaction, demonstrating the evolution from Holstein to Fr\"ohlich bipolarons and we compare the properties of bipolarons with singlet and triplet pairing. Finally, we present simulations of the bipolaron dispersion. The bandwidth of the Fr\"ohlich bipolaron is found to be broad and the decrease in bandwidth as the two polarons bind into a bipolaron is found to be far less rapid than in the case of the Holstein interaction. The properties of bipolarons formed from long-range electron-phonon interactions, such as light strongly bound bipolarons and intersite pairing when Coulomb repulsion is large, are found to be robust against screening, with qualitative differences between Holstein and screened Fr\"ohlich bipolarons found even for interactions screened within a single lattice site.
A theory of bipolaron states in quantum wires with a parabolic potential well is developed applying the Feynman variational principle. The basic parameters of the bipolaron ground state (the binding energy, the number of phonons in the bipolaron cloud, the effective mass, and the bipolaron radius) are studied as a function of sizes of the potential well. Two cases are considered in detail: a cylindrical quantum wire and a planar quantum wire. Analytical expressions for the bipolaron parameters are obtained at large and small sizes of the quantum well. It is shown that at $R\ensuremath{\gg}1$ [where
R means the radius (half width) of a cylindrical (planar) quantum wire, expressed in Feynman units], the influence of confinement on the bipolaron binding energy is described by the function $\ensuremath{\sim}{1/R}^{2}$ for both cases, while at small sizes this influence is different in each case. In quantum wires, the bipolaron binding energy
W(R) increases logarithmically with decreasing radius. The shapes and the sizes of a nanostructure, which are favorable for observation of stable bipolaron states, are determined.