A finite non-abelian $p$-group contains at least an element of order $p^2$

Question

Let $G$ be a finite non-abelian $p$-group such that $G$ contains at least an element of order $p^2$ and for every nontrivial normal subgroup $N$, $G/N$ has not any elements of order $p^2$ and $G/Z(G)$ is non-abelian group.

1. Is there a group with these properties?
2. Are there up to p non-trivial proper subgroups such that intersection of every non-trivial proper subgroup of G with at least one subgroup of these p subgroups is non-trivial.

Thank you in advance