The function $s \mapsto \epsilon(s, \pi)$ has the form $s \mapsto A e^{Bs}$ for some constants $A, B$, so it vanishes nowhere on $\mathbb{C}$. That has nothing to do with being a GJ lift, it's a general property of epsilon factors. Similarly, being a GJ lift doesn't really tell you much about the value at $s = \tfrac{1}{2}$: the standard L-function of a Gelbart--Jacquet lift doesn't look much different from any other automorphic representation of $GL(3)$.

What really separates the GJ lifts from other kinds of cuspidal auto reps of $GL(3)$ is that they are self-dual; so the Rankin--Selberg L-function of $\pi$ with itself, $L(\pi \times \pi, s)$, has a pole at $s = 1$ if and only if $\pi$ is a GJ lift (I hope I've remembered that correctly).

The function $s \mapsto \epsilon(s, \pi)$ has the form $s \mapsto A e^{Bs}$ for some constants $A, B$, so it vanishes nowhere on $\mathbb{C}$. That has nothing to do with being a GJ lift, it's a general property of epsilon factors. Similarly, being a GJ lift doesn't really tell you much about the value at $s = \tfrac{1}{2}$: analytically, the standard L-function of a Gelbart--Jacquet lift doesn't look much different from that of any other automorphic representation of $GL(3)$.

What really separates the GJ lifts from other kinds of cuspidal auto reps of $GL(3)$ is that they are self-dual; so the Rankin--Selberg L-function of $\pi$ with itself, $L(\pi \times \pi, s)$, has a pole at $s = 1$ if and only if $\pi$ is a GJ lift (I hope I've remembered that correctly). See also Peter Humphries' excellent answer to this question: How strong is the requirement of being a Gelbart-Jacquet lift?