A detailed understanding of inclusive muon neutrino charged-current interactions on argon is crucial to the study of neutrino oscillations in current and future experiments using liquid argon time projection chambers. To that end, we report a comprehensive set of differential cross section measurements for this channel that simultaneously probe the leptonic and hadronic systems by dividing the channel into final states with and without protons. Measurements of the proton kinematics and proton multiplicity of the final state are also presented. For these measurements, we utilize data collected with the MicroBooNE detector from 6.4$\times10^{20}$ protons on target from the Fermilab Booster Neutrino Beam at a mean neutrino energy of approximately 0.8 GeV. We present in detail the cross section extraction procedure, including the unfolding, and model validation that uses data to model comparisons and the conditional constraint formalism to detect mismodeling that may introduce biases to extracted cross sections that are larger than their uncertainties. The validation exposes insufficiencies in the overall model, motivating the inclusion of an additional data-driven reweighting systematic to ensure the accuracy of the unfolding. The extracted results are compared to a number of event generators and their performance is discussed with a focus on the regions of phase-space that indicate the greatest need for modeling improvements.