We report the first double-differential neutrino-argon cross section measurement made simultaneously for final states with and without protons for the inclusive muon neutrino charged-current interaction channel. The proton kinematics of this channel are further explored with a differential cross section measurement as a function of the leading proton's kinetic energy that extends across the detection threshold. These measurements utilize data collected using the MicroBooNE detector from 6.4$\times10^{20}$ protons on target from the Fermilab Booster Neutrino Beam with a mean neutrino energy of $\sim$0.8 GeV. Extensive data-driven model validation utilizing the conditional constraint formalism is employed. This motivates enlarging the uncertainties with an empirical reweighting approach to minimize the possibility of extracting biased cross section results. The extracted nominal flux-averaged cross sections are compared to widely used event generator predictions revealing severe mismodeling of final states without protons for muon neutrino charged-current interactions, possibly from insufficient treatment of final state interactions. These measurements provide a wealth of new information useful for improving event generators which will enhance the sensitivity of precision measurements in neutrino experiments.