  Cygnus X-3 (Cyg X-3) is a remarkable Galactic microquasar (X-ray binary)
emitting from radio to $\gamma$-ray energies. In this paper, we consider
hadronic model of emission of $\gamma$-rays above 100 MeV and their
implications. We focus here on the joint $\gamma$-ray and neutrino production
resulting from proton-proton interactions within the binary system. We find
that the required proton injection kinetic power, necessary to explain the
$\gamma$-ray flux observed by AGILE and Fermi-LAT, is $L_p \sim
10^{38}\:\rm{erg\:s^{-1}}$, a value in agreement with the average bolometric
luminosity of the hypersoft state (when Cygnus X-3 was repeatedly observed to
produce transient $\gamma$-ray activity). If we assume an increase of the wind
density at the superior conjunction, the asymmetric production of $\gamma$-rays
along the orbit can reproduce the observed modulation. According to
observational constraints and our modelling, a maximal flux of high-energy
neutrinos would be produced for an initial proton distribution with a power-law
index $\alpha=2.4$. The predicted neutrino flux is almost two orders of
magnitude less than the 2-month IceCube sensitivity at $\sim$1 TeV. If the
protons are accelerated up to PeV energies, the predicted neutrino flux for a
prolonged "soft X-ray state" would be a factor of about 3 lower than the 1-year
IceCube sensitivity at $\sim$10 TeV. This study shows that, for a prolonged
soft state (as observed in 2006) possibly related with $\gamma$-ray activity
and a hard distribution of injected protons, Cygnus X-3 might be close to being
detectable by cubic-kilometer neutrino telescopes such as IceCube.
