For Questions 1, 2. 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 $(d+1)\times (d+1)$ 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.