An empty set is always a subset of another set, but can it also be an element of a set too? I know an empty set ∅ is defined as a set that has no elements and is also a subset of every set, but can you say that it is an element of another set? I think not, but I'm not comepletly sure.
For example, consider the statements below:
$∅ ∈ P(Z)$
$∅ ⊆ Z$
$∅ ⊆ P(Z)$
$∅ ∈ Z$
I think only the second and third are correct since an empty set is always a subset of another set, but the first and last I'm not sure about. Would they be true statements or no? Also how do power sets affect this?

Actually yes, any set can be put as element of other sets.
This is ensured for example by the axiom of pairing in the axioms of the ZF set theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms.
You can for example have $Z=\{\emptyset\}$, and then $\emptyset\in Z$.
