We study the inverse Compton scattering of solar photons by Galactic cosmic-ray electrons. We show that the g-ray emission from this process is substantial, with the maximum flux in the direction of the Sun; the angular distribution of the emission is broad. This previously neglected foreground should be taken into account in studies of the diffuse Galactic and extragalactic g-ray emission. Furthermore, observations by GLAST can be used to monitor the heliosphere and determine the electron spectrum as a function of position from distances as large as Saturn's orbit to close proximity of the Sun, thus enabling unique studies of solar modulation. This paves the way for the determination of other Galactic cosmic-ray species, primarily protons, near the solar surface, which will lead to accurate predictions of g-rays from p-p interactions in the solar atmosphere. These albedo g-rays will be observable by GLAST, allowing the study of deep atmospheric layers, magnetic field(s), and cosmic-ray cascade development. The latter is necessary to calculate the neutrino flux from p-p interactions at higher energies (>1 TeV). Although this flux is small, it is a "guaranteed flux" in contrast to other astrophysical sources of neutrinos and may be detectable by km super(3) neutrino telescopes of the near future, such as IceCube. Since the solar core is opaque for very high energy neutrinos, directly studying the mass distribution of the solar core may thus be possible.