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Equilibrium topological phases are robust against weak static disorder but may break down in the strong-disorder regime. Here we propose to explore how the quench-induced emergent dynamical topology evolves under dynamical noise and uncover novel dynamical topological physics beyond equilibrium counterparts. We develop an analytic theory and show that for weak noise, the quantum dynamics induced by quenching an initial trivial phase to the Chern insulating regime exhibits robust emergent topology on certain momentum subspaces called band-inversion surfaces (BISs). The dynamical topology is protected by the minimal oscillation frequency over BISs, mimicking a bulk gap of the dynamical phase. Two novel types of dynamical transitions classified by distinct exceptional points or rings are predicted if increasing the noise to critical strength, with critical points being exactly obtained. At the exceptional points on the BISs the minimal oscillation frequency vanishes, manifesting the dynamical bulk-gap closing, beyond which the dynamical topology breaks down. Interestingly, we predict a sweet spot region of the transition, in which the dynamical topology survives surprisingly at an arbitrarily strong noise regime. This work unveils novel universal physics of dynamical topology under dynamical noise, which can be probed with control in experiment.