$p$-groups in which the centralizers are normal

For $(a)$, what you need to show is something more general: any subgroup of index $p$ in a $p$-group $G$ is a normal subgroup of $G$.

For $(b)$ remark that $G/C_G(x)$ is abelian for all $x$ thus

$G'\leq C_G(x)$ (by definition $G'$ is contained in every normal subgroup $N$ of $G$ such that $G/N$ is abelian).

And then use the equality :

$\bigcap_{x\in G}C_G(x)=Z(G)$.

For $(c)$, the Frattini subgroup is defined as the intersection of all maximal subgroups. Maybe you should stay simple. Although I quite don't see why, I would be inclined to think that you want to show that either $G=Z(G)$ or $|Z(G)|=p$.