  (Abridged) We study the predictions of various annihilating Dark Matter (DM)
models in order to interpret the origin of non-thermal phenomena in galaxy
clusters. We consider three neutralino DM models with light (9 GeV),
intermediate (60 GeV) and high (500 GeV) mass. The secondary particles created
by neutralino annihilation produce a multi-frequency Spectral Energy
Distribution (SED), as well as heating of the intracluster gas, that are tested
against the observations available for the Coma cluster. The DM produced SEDs
are normalized to the Coma radio halo spectrum. We find that it is not possible
to interpret all non-thermal phenomena observed in Coma in terms of DM
annihilation. The DM model with 9 GeV mass produces too small power at all
frequencies, while the DM model with 500 GeV produces a large excess power at
all frequencies. The DM model with 60 GeV and $\tau^{\pm}$ composition is
consistent with the HXR and gamma-ray data but fails to reproduce the EUV and
soft X-ray data. The DM model with 60 GeV and $b{\bar b}$ composition is always
below the observed fluxes. The radio halo spectrum of Coma is well fitted only
in the $b{\bar b}$ or light and intermediate mass DM models. The heating
produced by DM annihilation in the center of Coma is always larger than the
intracluster gas cooling rate for an NFW DM density profile and it is
substantially smaller than the cooling rate only for a cored DM density profile
in DM model with 9 GeV. We conclude that the possibility of interpreting the
origin of non-thermal phenomena in galaxy clusters with DM annihilation models
requires a low neutralino mass and a cored DM density profile. If we then
consider the multimessenger constraints to the neutralino annihilation
cross-section, it turns out that such scenario would also be excluded unless we
introduce a substantial boost factor due to the presence of DM substructures.
