In this paper, we study the random field solution to the stochastic nonlinear wave equation (SNLW) with constant initial conditions and multiplicative noise $σ(u)\dot{L}$, where the nonlinearity is encoded in a Lipschitz function $σ: \mathbb{R}\to\mathbb{R}$ and $\dot{L}$ denotes a pure-jump Lévy white noise on $\mathbb{R}_+\times\mathbb{R}$ with finite variance. Combining tools from Itô calculus and Malliavin calculus, we are able to establish the Malliavin differentiability of the solution with sharp moment bounds for the Malliavin derivatives. As an easy consequence, we obtain the spatial ergodicity of the solution to SNLW that leads to a law of large number result for the spatial integrals of the solution over $[-R, R]$ as $R\to\infty$. One of the main results of this paper is the obtention of the corresponding Gaussian fluctuation with rate of convergence in Wasserstein distance. To achieve this goal, we adapt the discrete Malliavin-Stein bound from Peccati, Solé, Taqqu, and Utzet ({\it Ann. Probab.}, 2010), and further combine it with the aforementioned moment bounds of Malliavin derivatives and Itô tools. Our work substantially improves our previous results (\textit{Trans.~Amer.~Math.~Soc.}, 2024) on the linear equation that heavily relied on the explicit chaos expansion of the solution. In current work, we also establish a functional version, an almost sure version of the central limit theorems, and the (quantitative) asymptotic independence of spatial integrals from the solution. The asymptotic independence result is established based on an observation of L. Pimentel (\textit{Ann.~Probab.}, 2022) and a further adaptation of Tudor's generalization (\textit{Trans.~Amer.~Math.~Soc.}, 2025) to the Poisson setting.