Philip Hall, the creator of modern $p$-group theory, has wrote only three papers devoted to general theory of finite $p$-groups. Essential part of their contents is not appeared in existing books.

I'd like to mention Chapters 3 and 8 in excellent Huppert's `Endliche Gruppen, I'. Their contents is not covered by mentioned above books.

Stanley has proved that the number of nonlinear irreducibles for a nonabelian $p$-group is a multiple of $p-1$. But Mann asserts more: the number of irreducibles of a given degree $>1$ is a multiple of $p-1$.

What you mean by `irreducibles generating the regular character'. The terminilogy of representation theory is well established, and there is no such notion. Hence, you have to give definition.

It is well known that if an abelian group $G=P_1\times\dots\times P_k$, where $P_i\in\text{Syl}(G)$, and $\text{d}(P_1)\ge\dots\ge\text{d}(P_k)$, then the minimal number of generators of $G$ is equal to $\text{d}(P_1)$. This is also true for nilpotent groups $G$.

Theorem 1 is stated non-correct. Should be: If $K$ is a $G$-class, $|K|>1$ and $|K|$ is a prime power, then $G$ is not simple. Lev Kazarin has proved that, in the case under consideration, $K$ is contained in the solvable radical of $G$.

Our group $G$ is supersolvable. So, by Wendt's theorem, $G'$ is nilpotent so cyclic. In that case, at least one of groups $G'$, $G/G'$ is of composite order. This is also true provided all Sylow subgroups of $G$ are cyclic and $|G|$ is a product of $>2$ primes.

2. Such classification exists only for $k=1$. For $k=2$, without additional assumptions (for example, provided exponent is equal to $p$) such classification is imposssible in view of variety of such groups,