We study the sensitivity of large-scale xenon detectors to low-energy solar neutrinos, to coherent neutrino-nucleus scattering and to neutrinoless double beta decay. As a concrete example, we consider the xenon part of the proposed DARWIN (Dark Matter WIMP Search with Noble Liquids) experiment. We perform detailed Monte Carlo simulations of the expected backgrounds, considering realistic energy resolutions and thresholds in the detector. In a low-energy window of 2-30 keV, where the sensitivity to solar pp and 7Be-neutrinos is highest, an integrated pp-neutrino rate of 5900 events can be reached in a fiducial mass of 14 tons of natural xenon, after 5 years of data. The pp-neutrino flux could thus be measured with a statistical uncertainty around 1%, reaching the precision of solar model predictions. These low-energy solar neutrinos will be the limiting background to the dark matter search channel for WIMP-nucleon cross sections below ~ 2 × 10-48 cm2 and WIMP masses around 50 GeVṡc-2, for an assumed 99.5% rejection of electronic recoils due to elastic neutrino-electron scatters. Nuclear recoils from coherent scattering of solar neutrinos will limit the sensitivity to WIMP masses below ~ 6 GeVṡc-2 to cross sections above ~ 4 × 10-45cm2. DARWIN could reach a competitive half-life sensitivity of 5.6 × 1026 y to the neutrinoless double beta decay of 136Xe after 5 years of data, using 6 tons of natural xenon in the central detector region.