1. The limiting function $V^\ast$ is such that any further convolutions of $e^{-V^\ast}$ with $\mu$ return $e^{-V^\ast}$, so $Z^\ast(\phi)=e^{-V^\ast(\phi)}$.

2. To obtain critical properties, you need the correlator $K(x,x')=\langle\phi(x)\phi(x')\rangle$. The decay length of the correlator diverges at the critical temperature $T_c$ as a power law $(T-T_c)^{-\alpha}$ and the power $\alpha$ is the critical exponent. The correlator is obtained by adding a source term $\lambda\psi(x)\psi(x')$ to the exponent in the definition of $Z(\phi)$ and then evaluating $dZ/d\lambda$ at $\phi=0$.

1. The limiting function $V^\ast$ is such that any further convolutions of $e^{-V^\ast}$ with $\mu$ return $e^{-V^\ast}$, so $Z^\ast(\phi)=e^{-V^\ast(\phi)}$.

2. To obtain critical properties, you need the correlator $K(x,x')=\langle\phi(x)\phi(x')\rangle$. The decay length of the correlator diverges at the critical temperature $T_c$ as a power law $(T-T_c)^{-\alpha}$ and the power $\alpha$ is the critical exponent. The correlator is obtained by adding a source term $\lambda\psi(x)\psi(x')$ to the exponent in the definition of $Z(\phi)$ and then evaluating $dZ/d\lambda$ at $\phi=0$.

In reference to the title of the post: "How can one recover/obtain information from the renormalization group procedure?" Information that depends on features that appear at small distances cannot be recovered, it is lost in the renormalization flow (which is not reversible). The information that remains refers to features that persist at large distances, such as a diverging correlation length and the critical exponents associated with it.