Ultrahigh-energy cosmic rays (UHECRs), with energy 1 EeV and above, propagate over cosmological distances, rendering them susceptible to interactions with the cosmic photon backgrounds that produce secondary particles, viz., neutrinos and gamma-rays. The sources, as well as the mass composition of UHECRs, can be constrained by probing these cosmogenic fluxes that extend to energies exceeding ~1 EeV. The neutrinos, being weakly interacting, travel unhindered and undeflected by extragalactic or Galactic magnetic fields. We fit the observed UHECR spectrum as measured by the Pierre Auger Observatory (PAO) by simulating the propagation of UHECRs of various mass compositions at injection, from different source distributions. We also calculate the cosmogenic neutrino spectrum for the same UHECR parameters, fitting the PAO data. The neutrino spectrum varies depending on the UHECR mass composition and source properties. Although the currently operating detectors do not reach the necessary sensitivity for observing cosmogenic neutrinos in all cases, a few parameter sets producing relatively high fluxes are already constrained by the flux upper limit imposed by 9-yrs of IceCube data. We also explore possibilities to constrain the UHECR abundance fraction of light nuclei at injection by identifying the fluxes of individual neutrino flavors in future detectors. Although the contribution of neutron beta decay to the cosmogenic neutrino spectrum is insignificant, it shifts the ratio of the fluxes of individual flavors obtained at earth from their constant values; thus serving as a discriminator between different mass composition models. The interactions leading to the production of neutrinos also leave their imprint on the observed UHECR spectrum. This can further explain the origin of cutoff in the UHECR spectrum at the highest energies.