Is there any length function on additive group of $\mathbb{Q}$ such that $\mathbb{Q}$ is of polynomial growth WRT this length function? What about the multiplicative group of $\mathbb{Q}$ instead?

Yes for $\mathbf{Q}$, no for $\mathbf{Q}^*$. For $\mathbf{Q}$, write it as the union of an increasing sequence $L_n$ with $L_1=\mathbf{Z}$ and $L_n$ of finite index over $L_1$. Pick a function $F$ with fast growth and define $l'(r)=r+F(\sup\{n:r\notin L_n\})$. For $\mathbf{Q}^*$, it contains a subgroup isomorphic to $\mathbf{Z}^d$ for every $d$, so for every length the growth is at least polynomial of degree $d$. So it can't be polynomial. Added1: for $\mathbf{Q}$ we can arrange so that $[L_n:L_1]\le n$ for all $n$. Then we can pick $F$ to be the identity and then the growth is at most quadratic. Added2: the argument extends, showing that an abelian group admits a length with polynomial growth iff it's countable and has finite $\mathbf{Q}$rank. 

