(a) Let $G$ have polynomial growth of degree d. Let the polynomial growth function of G under the generating set S given by $\gamma_S(n)\leq c_1n^d$. Does this imply that under any other generating set T, $\gamma_T(n)\leq c_2n^d$, possibly with $c_2 \neq c_1$?
Suppose not, then $$\gamma_S(n)\leq c_2n^d<\gamma_T(n)\leq c_1n^d$$ I'm trying to find something here that will lead to contradiction.
All I have from this is that for some fixed $n$, $\mid B_S(n)\mid\leq \mid B_T(n)\mid$ where $B_S(n)$ is the ball of radius $n$ under $S$. This implies that $$\exists g\in G: g\in B_T(n) \text{ and } g\notin B_S(n)$$ Thus $g=t_1t_2...t_n$ for $t_i\in T$, but $g$ requires more than $n$ elements to be described as a word in $S$.
I also know that $$l_S(g)\leq max\{l_S(y):y\in T\}*l_T(g)=max\{l_S(y):y\in T\}*n$$ where $l_s(g)$ is the length of a word with respect to $S$. Perhaps this can lead to a contradiction since it puts a bound on the length of $g$ w.r.t $S$. But I don't see one.
Let $T$ be any other finite generating subset. There exists some integer $N$ such that every $g \in S$ is the product of at most $N$ elements of $T$ and vice-versa.
Then, for any integer $k \geq N^2$, $ c_Nk^d\leq B_S(\left\lfloor{k/N}\right \rfloor) \subset B_T(k) \subset B_S(kN) \leq C_Nk^d$ for constants $0 < c_N < C_N$.
That should give you the answer, I think.