I admit that I haven't read it carefully, but in this paper E. Kobayashi conjectures that $E(\mathbb Q^{\rm ab})$ has infinite rank for all elliptic curves $E$ defined over $\mathbb Q^{\rm ab}$. In particular, assuming the "weak" Birch and Swinnerton-Dyer conjecture for $E$ and certain properties of twisted Hasse-Weil $L$-functions of $E$, she shows that if $E$ is defined over a number field $\mathbb Q$ K$ of odd degree then $E(\mathbb E(K\cdot\mathbb Q^{\rm ab})$ has infinite rank (Theorem 2 in Kobayashi's article, which is proved, more generally, when $\mathbb Q$ is replaced by a number field $K$ of odd degree and $\mathbb Q^{\rm ab}$ is replaced by $K\cdot\mathbb Q^{\rm ab}$)article).
This result for elliptic curves seems to suggest that the answer to your question (a) might be "no".
As for a simple abelian variety $A$ over a number field $K$, it is perhaps worth pointing out that Zarhin proved here that the torsion subgroup of $A(K^{\rm ab})$ is finite if and only if $A$ is not of CM-type over $K$ (when $K=\mathbb Q$ this reduces to an earlier theorem of Ribet).
