About the existence of a generic point on an irreducible closed subset of a prescheme

About the existence of a generic point on an irreducible closed subset of
a prescheme

This is Proposition 2 on page 81 of Mumford's The Red Book of Varieties
and Schemes:
Let $X$ be a prescheme, and $Z \subset X$ an irreducible closed subset.
Then there is one and only one point $z \in Z$ such that $Z = \overline{\{
z \}}$.
Proof. Let $U \subset X$ be an open affine set such that $Z \cap U \neq
\emptyset$. Then any point $z \in Z$ dense in $Z$ must be in $Z \cap U$;
and a point $z \in Z \cap U$ whose closure contains $Z \cap U$ is also
dense in $Z$. Therefore it suffices to prove the theorem for the closed
subset $Z \cap U$. But by Prop. 1 of section 4 there is a unique $z \in Z
\cap U$ dense in $Z \cap U$.
I have some questions about this proof.
Why does every dense point in $Z$ lie in $Z \cap U$?
Why must $z \in Z \cap U$ whose closure contains $Z \cap U$ be dense in $Z$?
Why is there a unique $z \in Z \cap U$ dense in $Z \cap U$? I can't find
the proposition the author mentioned.
I think these are concerned with affineness, but I don't know the exact
reason.
Thanks for everyone.