Pursuing complementary field-theoretic and numerical methods, we here paint the global phase diagram of a three-dimensional dirty Weyl system. The generalized Harris criterion, augmented by a perturbative renormalization-group analysis shows that weak disorder is an irrelevant perturbation at the Weyl semimetal (WSM)-insulator quantum-critical point. But, a metallic phase sets in through a quantum phase transition (QPT) at strong disorder across a multicritical point. The field-theoretic predictions for the correlation length exponent v = 2 and dynamic scaling exponent z = 5/4 at this multicritical point are in good agreement with the ones extracted numerically, yielding v = 1.98 +/- 0.10 and z = 1.26 +/- 0.05, from the scaling of the average density of states (DOS). Deep inside the WSM phase, generic disorder is also an irrelevant perturbation, while a metallic phase appears at strong disorder through a QPT. We here demonstrate that in the presence of generic but strong disorder, the WSM-metal QPT is ultimately always characterized by the exponents v = 1 and z = 3/2 (to one-loop order), originating from intranode or chiral-symmetric (e.g., regular and axial potential) disorder. We here anchor such emergent chiral super-universality through complementary renormalization-group calculations, controlled via. expansions, and numerical analysis of average DOS across WSM-metal QPT. In addition, we also discuss a subsequent QPT (at even stronger disorder) of a Weyl metal into an Anderson insulator by numerically computing the typical DOS at zero energy. The scaling behavior of various physical observables, such as residue of quasiparticle pole, dynamic conductivity, specific heat, Gruneisen ratio, inside various phases as well as across various QPTs in the global phase diagram of a dirty Weyl liquid, are discussed.