  Galaxy Clusters (GCs) are the largest reservoirs of both dark matter and
cosmic rays (CRs). Dark matter self-annihilation can lead to a high luminosity
in gamma rays and neutrinos, enhanced by a strong degree of clustering in dark
matter substructures. Hadronic CR interactions can also lead to a high
luminosity in gamma rays and neutrinos, enhanced by the confinement of CRs from
cluster accretion/merger shocks and active galactic nuclei. We show that
IceCube/KM3Net observations of high-energy neutrinos can probe the nature of
GCs and the separate dark matter and CR emission processes, taking into account
how the results depend on the still-substantial uncertainties. Neutrino
observations are relevant at high energies, especially at >10 TeV. Our results
should be useful for improving experimental searches for high-energy neutrino
emission. Neutrino telescopes are sensitive to extended sources formed by dark
matter substructures and CRs distributed over large scales. Recent observations
by Fermi and imaging atmospheric Cherenkov telescopes have placed interesting
constraints on the gamma-ray emission from GCs. We also provide calculations of
the gamma-ray fluxes, taking into account electromagnetic cascades inside GCs,
which can be important for injections at sufficiently high energies. This also
allows us to extend previous gamma-ray constraints to very high dark matter
masses and significant CR injections at very high energies. Using both
neutrinos and gamma rays, which can lead to comparable constraints, will allow
more complete understandings of GCs. Neutrinos are essential for some dark
matter annihilation channels, and for hadronic instead of electronic CRs. Our
results suggest that the multi-messenger observations of GCs will be able to
give useful constraints on specific models of dark matter and CRs. [Abstract
abridged.]
