The first three statements are true for $n$ sufficiently large.

Regarding a), a classical result of Ingham (1937) shows that
$$p_n^{1/3} - p_{n-1}^{1/3}<\frac{p_n-p_{n-1}}{3p_{n-1}^{2/3}}=o(1).$$

Regarding b), the Prime Number Theorem gives that
$$p_n^{1/n} - p_{n-1}^{1/n}<\frac{p_n-p_{n-1}}{np_{n-1}^{(n-1)/n}}\sim\frac{p_n-p_{n-1}}{n p_{n-1}}=o\left(\frac{1}{n}\right).$$

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 the left-hand side tends to zero (as $n\to\infty$).