Yes, this is a corollary of the following famous theorem of Marshall Hall:

Theorem: If $A$ is a finitely generated subgroup of a free group $G$, then there exists a finite-index subgroup $H$ of $G$ containing $A$ such that $A$ is a free factor of $H$, i.e. such that $G = A \ast A'$ for some subgroup $A'$ of $G$.

Just apply this to the cyclic subgroup $A = \langle g \rangle$.

For a proof of Marshall Hall's theorem, see e.g. this blog post.

Yes, this is a corollary of the following famous theorem of Marshall Hall:

Theorem: If $A$ is a finitely generated subgroup of a free group $G$, then there exists a finite-index subgroup $H$ of $G$ containing $A$ such that $A$ is a free factor of $H$, i.e. such that $H = A \ast A'$ for some subgroup $A'$ of $G$.

Just apply this to the cyclic subgroup $A = \langle g \rangle$.

For a proof of Marshall Hall's theorem, see e.g. this blog post.