  Suppose $A$ is a $C^*$-algebra and $H$ is a $C^*$-correspondence over $A$. If
$H$ is regular in the sense that the left action of $A$ is faithful and is
given by compact operators, then we compute the $K$-theory of $\mathcal{O}_A(H)
\rtimes \mathbb{T}$ where the action is the usual gauge action. The case where
$A$ is an AF-algebra is carefully analyzed. In particular, if $A$ is AF, we
show $\mathcal{O}_A(H) \rtimes \mathbb{T}$ is AF. Combining this with Takai
duality and an AF-embedding theorem of N. Brown, we show the conditions
AF-embeddability, quasidiagonality, and stable finiteness are equivalent for
$\mathcal{O}_A(H)$. If $H$ is also assumed to be regular, these finiteness
conditions can be characterized in terms of the ordered $K$-theory of $A$.
