Detection of the IceCube-170922A neutrino coincident with the flaring blazar TXS 0506+056, the first and only ∼3 high-energy neutrino source association to date, offers a potential breakthrough in our understanding of high-energy cosmic particles and blazar physics. We present a comprehensive analysis of TXS 0506+056 during its flaring state, using newly collected Swift, NuSTAR, and X-shooter data with Fermi observations and numerical models to constrain the blazar's particle acceleration processes and multimessenger (electromagnetic (EM) and high-energy neutrino) emissions. Accounting properly for EM cascades in the emission region, we find a physically consistent picture only within a hybrid leptonic scenario, with γ-rays produced by external inverse-Compton processes and high-energy neutrinos via a radiatively subdominant hadronic component. We derive robust constraints on the blazar's neutrino and cosmic-ray emissions and demonstrate that, because of cascade effects, the 0.1-100 keV emissions of TXS 0506+056 serve as a better probe of its hadronic acceleration and high-energy neutrino production processes than its GeV-TeV emissions. If the IceCube neutrino association holds, physical conditions in the TXS 0506+056 jet must be close to optimal for high-energy neutrino production, and are not favorable for ultrahigh-energy cosmic-ray acceleration. Alternatively, the challenges we identify in generating a significant rate of IceCube neutrino detections from TXS 0506+056 may disfavor single-zone models, in which γ-rays and high-energy neutrinos are produced in a single emission region. In concert with continued operations of the high-energy neutrino observatories, we advocate regular X-ray monitoring of TXS 0506+056 and other blazars in order to test single-zone blazar emission models, clarify the nature and extent of their hadronic acceleration processes, and carry out the most sensitive possible search for additional multimessenger sources.