A full energy and flavor-dependent analysis of the three-year high-energy IceCube neutrino events is presented. By means of multidimensional fits, we derive the current preferred values of the high-energy neutrino flavor ratios, the normalization and spectral index of the astrophysical fluxes, and the expected atmospheric background events, including a prompt component. A crucial assumption resides on the choice of the energy interval used for the analyses, which significantly biases the results. When restricting ourselves to the similar to 30 TeV-3 PeV energy range, which contains all the observed IceCube events, we find that the inclusion of the spectral information improves the fit to the canonical flavor composition at Earth, (1: 1: 1)(circle plus), with respect to a single-energy bin analysis. Increasing both the minimum and the maximum deposited energies has dramatic effects on the reconstructed flavor ratios as well as on the spectral index. Imposing a higher threshold of 60 TeV yields a slightly harder spectrum by allowing a larger muon neutrino component, since above this energy most atmospheric tracklike events are effectively removed. Extending the high-energy cutoff to fully cover the Glashow resonance region leads to a softer spectrum and a preference for tau neutrino dominance, as none of the expected electron (anti) neutrino induced showers have been observed so far. The lack of showers at energies above 2 PeV may point to a broken power-law neutrino spectrum. Future data may confirm or falsify whether the recently discovered high-energy neutrino fluxes and the long-standing detected cosmic rays have a common origin.