I was wondering what the difference between these 2 predicate statements were and if they were equivalent or not.
$$ \forall x\in \Bbb N \ \exists y \in \Bbb N: \ \neg P(x) \implies P(y) \ \land Q(x,y) $$
and
$$ \forall x\in \Bbb N :\ \neg P(x) \implies \ \exists y \in \Bbb N :\ P(y) \land Q(x,y) $$
Those are equivalent.
It is an instance of one of the Prenex Laws that state how you can move quantifiers around in a predicate logic statement.
In this case, the relevant Prenex Law is:
$\psi \to \exists x \varphi(x) \Leftrightarrow \exists x (\psi \to \varphi(x))$
where $\varphi(x)$ is any formula that has $x$ as a free variable, but with $\psi$ as a formula that does not have any free variables $x$.
In your case, we have that $\neg P(x)$ does not have any free variable $y$, and hence we can apply this Prenex Law to show that
$$\exists y \in \Bbb N: \ \neg P(x) \implies P(y) \ \land Q(x,y)$$
and
$$\neg P(x) \implies \ \exists y \in \Bbb N :\ P(y) \land Q(x,y)$$
are equivalent, meaning that
$$ \forall x\in \Bbb N \ \exists y \in \Bbb N: \ \neg P(x) \implies P(y) \ \land Q(x,y) $$
and
$$ \forall x\in \Bbb N :\ \neg P(x) \implies \ \exists y \in \Bbb N :\ P(y) \land Q(x,y) $$
are equivalent as well.