The answer is "no" in both cases.

The terms of the lower central series of a group are verbal subgroups. If we let $\gamma_c(G)$ denote the $c$th term of the lower central series of $G$, then for any groups $G$ and $K$ and any group homomorphism $\varphi\colon G\to K$, we have $\varphi(\gamma_c(G))=\gamma_c(\varphi(G))\subseteq \gamma_c(K)$; and if $\varphi$ is onto, then $\varphi(\gamma_c(G))$ maps onto $\gamma_c(K)$, so that we have equality.

In particular, if $\pi\colon G\to G/N$ is a quotient map, then $\gamma_c(G/N)$ is trivial if and only if $\gamma_c(G)\subseteq N$. That is, the quotient $G/N$ is nilpotent of class at most $c-1$ if and only if the $c$th term of the lower central series of $G$ is contained in $N$.

So let $F$ be a free group, and let $N$ be a normal subgroup of $F$. Then $N$ contains $\gamma_c(F)$ if and only if $F/N$ is nilpotent of class at most $c-1$.

So, for example, if $F$ is the free group on two generators, then there is a normal subgroup of $F$ such that $F/N\cong S_3$; since $S_3$ is not nilpotent, $N$ does not contain any term of the lower central series of $F$, even though $N$ is of finite index.