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In this work, we present the results of searches for signatures of dark matter decay or annihilation into Standard Model particles, and secret neutrino interactions with dark matter. Neutrinos could be produced in the decay or annihilation of galactic or extragalactic dark matter. Additionally, if an interaction between dark matter and neutrinos exists then dark matter will interact with extragalactic neutrinos. In particular galactic dark matter will induce an anisotropy in the neutrino sky if this interaction is present. We use seven and a half years of the High-Energy Starting Event (HESE) sample data, which measures neutrinos in the energy range of approximately 60 TeV to 10 PeV, to study these phenomena. This all-sky event selection is dominated by extragalactic neutrinos. For dark matter of similar to 1 PeV in mass, we constrain the velocity-averaged annihilation cross section to be smaller than 10(-23) cm(3)/s for the exclusive mu(+) mu(-) channel and 10(-22) cm(3)/s for the b (b) over bar channel. For the same mass, we constrain the lifetime of dark matter to be larger than 10(28) s for all channels studied, except for decaying exclusively to b (b) over bar where it is bounded to be larger than 10(27) s. Finally, we also search for evidence of astrophysical neutrinos scattering on galactic dark matter in two scenarios. For fermionic dark matter with a vector mediator, we constrain the dimensionless coupling associated with this interaction to be less than 0.1 for dark matter mass of 0.1 GeV and a mediator mass of 10(-4) GeV. In the case of scalar dark matter with a fermionic mediator, we constrain the coupling to be less than 0.1 for dark matter and mediator masses below 1MeV.