IceCube is a cubic-kilometer neutrino telescope nearing completion in the South Pole Ice. Designed to detect astrophysical neutrinos from 100 GeV to about an EeV, it will contribute to the fields of high energy astrophysics, particle physics, and neutrino physics. This analysis looks at the flux of atmospheric neutrinos detected by IceCube while it operated in a partially-completed, 40-string configuration, from April 2008 to May 2009. From this data set, a sample of about 20,000 up-going atmospheric muon neutrino events with negligible background was extracted using Boosted Decision Trees. A discrete Fourier transform method was used to constrain a directional asymmetry in right ascension. Constraints on certain interaction coefficients from the Standard Model Extension were improved by three orders of magnitude, relative to prior experiments. The event sample was also used to unfold the atmospheric neutrino spectrum at its point of origin, and seasonal and systematic variations in the atmospheric muon neutrino flux were studied. A likelihood method was developed to constrain perturbations to the energy and zenith angle dependence of the atmospheric muon neutrino flux that could be due to Lorentz-violating oscillations or decoherence of neutrino flavor. Such deviations could be a signature of quantum gravity in the neutrino sector. The impact of systematic uncertainties in the neutrino flux and in the detector response on such a likelihood analysis were examined. Systematic uncertainties that need to be reduced in order to use a two-dimensional likelihood analysis to constrain phenomenological models for Lorentz or CPT violating neutrino oscillations were identified.