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$this reduces to an earlier theorem of Ribet). 1 I admit that I haven't read it carefully, but in this paper E. Kobayashi conjectures that$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$is defined over$\mathbb Q$then$E(\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}$). This result for elliptic curves seems to suggest that the answer to your question (a) might be "no". As for a simple abelian variety$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).