For a group $G$, its FC-Center $FC(G)$ is the subgroup consisting of the elements of $G$ which have a finite conjugacy class (in $G$).
Let $G$ be a finitely generated group. Is $FC(G)$ necessarily finitely generated as well?
In case the answer is no, what if we assume further that $G$ is finitely presented?
The example I gave in my answer to Center of a finitely generated group is a finitely generated group in which the centre is not finitely generated. In fact $Z(G) = {\rm FC}(G)$ in that example, so the answer to your first question is no.
In a comment, Arturo Magidin gave a reference for an example of a finitely presented group in which $Z(G)$ is not finitely generated. You would need to look at the example and check whether it also has ${\rm FC}(G)$ not finitely generated. Perhaps $Z(G)={\rm FC}(G)$ in that example too, but I don't know!