How does one prove that an Eisenstein series (adelically formulated as in the book of MoeglinWaldspurger) is not identically zero? Namely how does one prove that the sum $\sum_{\gamma\in P(k)\backslash G(k)}\varphi(\gamma g)$ is not identically zero, provided the sum converges? The book by MW does not prove it, and neither could I find a reference.

The space $P(k) \backslash G(k)$ has a discrete set of representatives in $ U(A) M(k) \backslash G(A)$. See for example the discussion after Lemma 3.3 in GelbartJacquet "Forms of GL(2) from the analytic view point" for $G=GL(2)$ . I am sure somewhere in MoeglinWaldspurger a similar lemma is quoted/proved somewhere for the more general $G$. Arthur has something similar certainly for $G$ reductive, but I remember that MW consider also more generally finite covers etc. The main concern of these lemmas is actually the absolute convergence, but also they also provide the nontriviality. A suitable set of representatives can be given via the Bruhat decomposition. Pick a compact set $K \subset U(A) M(k) \backslash G(A)$ containing only one representative and having nonempty interior. Let $\varphi :U(A) M(k) \backslash G(A) \rightarrow \mathbb{C}$ be a function which is positive, smooth, compactly supported on this set $K$, nonvanishing on the interior. Then your sum will not be zero only for $\gamma^{1} g$ for $ g \in K$. This works equally well if you work modulo the center. 

