We study the fractional $Φ^4_3$-measure (with order $α> 1$) and the dynamical problem of its canonical stochastic quantization: the three-dimensional stochastic damped fractional nonlinear wave equation with a cubic nonlinearity, also called the fractional hyperbolic $Φ^4_3$-model. We first construct the fractional $Φ^4_3$-measure via the variational approach by Barashkov-Gubinelli (2020). When $α\leq \frac{9}{8}$, this fractional $Φ^4_3$-measure turns out to be singular with respect to the base Gaussian measure. We then prove almost sure global well-posedness of the fractional hyperbolic $Φ^4_3$-model and invariance of the fractional $Φ^4_3$-measure for all $α> 1$ by further developing the globalization framework due to Oh-Okamoto-Tolomeo (2024) on the hyperbolic $Φ^3_3$-model. Furthermore, when $α> \frac{9}{8}$ , we prove weak universality of the fractional hyperbolic $Φ^4_3$-model by utilizing the convergence of Gibbs measures.