Regarding a), a classical result of Ingham (1937) shows that for every sufficiently large $n$ we have $$p_n^{1/3} - p_{n-1}^{1/3}<\frac{p_n-p_{n-1}}{3p_{n-1}^{2/3}}<\frac{1}{3}.$$ In fact Ingham's bound implies that the fraction tends to zero (as $n\to\infty$), hence so does the left-hand side.
Regarding c), Bertrand's postulate shows that for $p_n>7$ we have $$(\log p_n)^{1/2}-(\log p_{n-1})^{1/2}<\frac{\log p_n-\log p_{n-1}}{2(\log p_{n-1})^{1/2}}<\frac{\log 2}{2(\log p_{n-1})^{1/2}}<\frac{1}{4}.$$ In fact we see again that the left-hand side tends to zero (as $n\to\infty$).
Regarding d), we have by standard estimates that $$(\log p_n)^{1/n}-(\log p_{n-1})^{1/n}\sim\frac{\log p_n-\log p_{n-1}}{n\log n}\sim\frac{1}{n^2\log n}.$$ Hence the left-hand side is less than $1/n^2$ for $n$ large, and the exponent $2$ cannot be increased. With little effort the assumption "$n$ large" can be made fully explicit.
I will write more as time permits.