No. Let $X$ be the affine plane with coordinates $(s,t)$. Let $Z$ be the affine plane with coordinates $(u,v)$. Let $Y$ be the affine line with coordinate $w$. Define $g$ by $(u,v)=(s,st)$ and define $f$ by $w=st^2$.

No. Consider the case in which $X$ is the disjoint union of a closed subset of $Z$ and the open complement, and $g:X\to Z$ is the obvious bijection. Let $Y$ be $X$ and let $f$ be the identity map.