This partial solution by Moshe Newman:
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.