Say, $B$ is a category fibered in groupoids over some category $C$, and $A$ is a category fibered in groupoids over $B$.

Suppose $A$ is a stack (over whatever site) over $B$, and $B$ is a stack over $C$, is $A$ also a stack over $C$?

"Conversely", if $A$ is a stack over $C$ and $B$ is a stack over $C$, then must $A$ be a stack over $B$?

Also, do properties like being algebraic, having a coarse moduli space, etc, depend on what you consider the base category to be? (Ie, if $A$ as a stack over $B$ is algebraic/has a CMS, then must $A$ as a stack over $C$ be algebraic/have a CMS as well?)