Subtle point in definition of BNS invariant

Let $S$ and $T$ be generating sets for $G$, and suppose that $X_\phi$ is $T$-connected (i.e., spans a connected subgraph in the Cayley graph of $G$ with respect to $T$.) Let $[X_\phi]_S$ be the subgraph of $\text{Cay}(G,S)$ spanned by $X_\phi$. We must show that $[X_\phi]_S$ is connected.

Claim 1: For any $n\in\mathbb{Z}$, the set $\{g\in G:\phi(g)\geq n\}$ is $T$-connected.

Proof: This follows, by translation, from the fact that $X_\phi$ is $T$-connected.

Claim 2: There exists some $n$ such that for any $g\in G$ and $t\in T$, there is a path in $\text{Cay}(G,S)$ from $g$ to $gt$ such that if $v$ is a vertex of this path, then $\phi(v)>\phi(g)-n$ and $\phi(v)>\phi(gt)-n$.

Proof: For each $t\in T$, choose some word $w_t$ in the alphabet $S$ representing $t$. Choose $n$ so that it is larger than $|\phi(w)|$ whenever $w$ is a prefix or suffix of any $w_t$. The claim follows by connecting $g$ and $gt$ via the path given by $g \cdot w_t$.

If $n$ is as in Claim 2, it follows that any two points of $\{g\in G:\phi(g)\geq n\}$ may be connected by a path in $[X_\phi]_S$.

Claim 3: there is some $s \in S \cup S^{-1}$ such that $\phi(s) > 0$.

Proof: this follows from the fact that $\phi$ is not identically $0$, by definition.

By Claim 3 we see that, for any $g \in X_\phi$, there is a path $g,gs,gs^2,\ldots$ in $[X_\phi]_S$ from $g$ to $\{g\in G:\phi(g)\geq n\}$. Since any two vertices of $\{g\in G:\phi(g)\geq n\}$ may be connected by a path in $[X_\phi]_S$, it follows that $[X_\phi]_S$ is connected.

This is a reference of question 2.

Various definitions (including the original one and the one you have given) of BNS invariant are discussed in Chapter C of this lecture notes by Ralph Strebel who call the BNS invariant as "Sigma invariants".