We report on searches for neutrinos and antineutrinos from astrophysical sources performed with the Borexino detector at the Laboratori Nazionali del Gran Sasso in Italy. Electron antineutrinos (ν¯_e) are detected in an organic liquid scintillator through the inverse β-decay reaction. In the present work we set model-independent upper limits in the energy range 1.8–16.8 MeV on neutrino fluxes from unknown sources that improve our previous results, on average, by a factor 2.5. Using the same data set, we first obtain experimental constraints on the diffuse supernova ν¯_e fluxes in the previously unexplored region below 8 MeV. A search for ν¯_e in the solar neutrino flux is also presented: the presence of ν¯_e would be a manifestation of a non-zero anomalous magnetic moment of the neutrino, making possible its conversion to antineutrinos in the strong magnetic field of the Sun. We obtain a limit for a solar ν¯_e flux of 384 cm^–2 s^–1 (90% C.L.), assuming an undistorted solar ⁸B neutrinos energy spectrum, that corresponds to a transition probability p_νe→ν¯e< 7.2 × 10^–5 (90% C.L.) for E_ν¯e > 1.8 MeV. At lower energies, by investigating the spectral shape of elastic scattering events, we obtain a new limit on solar ⁷Be-ν_e conversion into ν¯_e of p_νe→ν¯e< 0.14 (90% C.L.) at 0.862 MeV. Last, we investigate solar flares as possible neutrino sources and obtain the strongest up-to-date limits on the fluence of neutrinos of all flavor neutrino below 3–7 MeV. Assuming the neutrino flux to be proportional to the flare's intensity, we exclude an intense solar flare as the cause of the observed excess of events in run 117 of the Cl-Ar Homestake experiment.