A module over an algebra. Is it a vector space?
Yes. A module $M$ over $A$ is just a module $M$ over the ring $A$; the additional structure of $A$ as a $k$-algebra plays no role.
(and 3.) That amounts to the same: since $A$ is a $k$-algebra, you already have a map $k \to A$ which turns $M$ into a $k$-module as well.
$M$, ultimately being a $k$-module as well, is a $k$-vector space. I'm not sure what you mean by "Is $M$ a vector space in the case of 1?"