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Let $G$ be a finitely generated group with polynomial growth rate. Denote the growth rate by $\Theta(n^k)$ for some specific set of generators.

Consider the sizes of spheres of a given radius $n$ (rather than considering balls as in the definition of growth rate). Is it then true that the size of a sphere is $O(n^{k-1})$?

It seems plausible, and this is what happens with free abelian groups, but theoretically it may be possible that some sphere of one specific radius is very large.

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You can find some information in this preprint by Breuillard and Le Donne. They say that this is an open problem in general, and for a nilpotent group of size $n$ they say that the growth is at most $O(n^{k-2/3r})$ where $r$ is the nilpotency length. They also say that the $O(n^{k-1})$ bound holds when $r=2$ by a result of Stoll (On the asymptotics of the growth of 2-step nilpotent groups. J. London Math. Soc. (2), 58(1):38–48, 1998). I don't know if the virtually nilpotent case boils down to the nilpotent case.

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