This partial solution by Stuart Margolis:
The answer to question $2$ is definitely yes, the number of elements required to generate $G$ can't be more than $D=2^{d-1}$. Suppose that $G$ requires more than $D$ generators, then every $D$-tuple of elements of $G$ must belong to a proper subgroup, which therefor must be solvable of derived length at most $d-1$. So every $D$-tuple of elements satisfies the identity (in $D$ variables) that defines being solvable of derived length at most $d-1$. So $G$ itself must be solvable of derived length at most $d-1$.
This is just a special example of the fact that if a variety (in the Hanna Neumann sense) is defined by an $n$-variable law then: a group which is not in the variety, but all its subgroups do belong to the variety, must be generated by at most $n$ elements.
