Nested quantifiers for $y < x^2$ 
*

*$∀x ∃y P(x, y)$
For every $x$ there exists a $y…$
This is true, because for every number $x$ there exists at least one number $y$ for the statement to be true. For example, we can choose $x=100$ and $y =10.$


*$∀y ∃x P(x, y)$
?


*$∃x ∀y P(x, y)$
There exists a $x$ (such that) for every $y…,$
This is true, we can increment $x$ and $y$ to infinite, and every
time the statement will stay true. If we set an end to the infinite
and give the largest number possible to $x,$ that number will hold
true for any possible $y.$


*$∃y ∀ x P(x, y)$
There exists a $y$ (such that) for every $x…,$ means that there is a
$y$ that is less than $x^2$ for every possible $x.$ This is true
because we can give $y$ a negative value, whatever value we will
give $x$ will still be more than any negative value because it is
quadratic.
Is this how you correctly understand quantifiers? Or am I understanding wrongly?
 A: Your understanding and examples for your first and last sentence sounds right.
As for the sentence $\forall y \exists x P(x, y)$ in your case, it reads "for any $y \in \mathbb{R}$ there's some $x \in \mathbb{R}$ such that $y<x^2$" which is certainly true as there's no upper bound of the term $x^2$.
As for the sentence $\exists x \forall y P(x, y)$ in your case, it reads "there's some $x \in \mathbb{R}$ for any $y \in \mathbb{R}$ it's the case that $y<x^2$" which is clearly false since if you happens to find such $x$, you can always find a positive $y>x^2$.
In conclusion order matters for mixed quantifiers with different types...
A: 
P.S. Forgot to add that x and y are REAL numbers

It isn't necessary to specify this, since complex numbers cannot be ordered anyway.


*

*$∀x ∃y P(x, y)$
For every $x$ there exists a $y…$
This is true, because...

Yes.



*$∀y ∃x P(x, y)$

This is true, because no matter what $y$ is, there is always some $x$ whose square is bigger, since $\mathbb R$ has no upper bound.



*$∃x ∀y P(x, y)$
There exists a $x$ (such that) for every $y…,$
This is true, we can increment $x$ and $y$ to infinite, and every time the statement will stay true. If we set an end to the infinite and give the largest number possible to $x,$ that number will hold true for any possible $y.$

Proposition $3$ is false, because no such $x$ exists, precisely because we cannot "set an end to the infinite".



*$∃y ∀ x P(x, y)$
There exists a $y$ (such that) for every $x…,$ means that the is a
$y$ that is less than $x^2$ for every possible $x.$ This is true
because...

Yes. But do be careful though: "there is a $y$ that is less than $x^2$ for every possible $x$" is ambiguous (it contains a hanging quantifier), and can be interpreted either as $∃y ∀x P(x, y)$ or $∀x ∃y P(x, y).$
