What roots of unity can be contained in the abelian extensions of an imaginary quadratic number field $K = \mathbb{Q}(\sqrt{-d})$? In particular, I would like to know:

Is $K(\zeta_n)/K$ an abelian extension for every $n$?

What are the roots of unity in the ray class field of $K$ with conductor $\mathfrak{c}$?

What are the roots of unity in the ring class field of the order $\mathcal{O} = \mathbb{Z} + f\mathcal{O}_K$ with conductor $f$?