For Question 1. 
We know $c(d,2)=1$ for all $d$ and $c(1,3)=1$, $c(2,3)=2$ and $c(d,3)=3$ for all $d>2$.
So suppose if necessary $p\geq 5$. 

Take $G(d,p)$ to be the unitiriangular matrices of size $d\times d$ over the field of size $p$, where $p>d$. Then $G(d,p)$ is nilpotent of class $d$ and the exponent is $p$. The group can be generated by $d$ elements.