First, I know that you know that the inference rules aren't fractions, but still ... please replace 'Numerator' and 'Denominator' with something more appropriate ... such as 'premise' and 'conclusion' respectively.
OK, the rules, and some more intuitive explanation:
Universal Instantiation
'Typical' Form:
$\forall x P(x)$
$\therefore P(a)$ for any constant $a$
Explanation:
I all things have property $P$, then of course each individual thing has property $P$, whether this is $a$, $b$, ... This is why there are no restrictions here.
Universal Generalization
'Typical' Form:
$P(a)$ ... where $a$ has been introduced as some arbitrary object!
$\therefore \forall x P(x)$
Explanation:
Suppose we have a constant that we are using to denote a specific object, e.g. suppose we use the constant $c$ for 'Charlie', and suppose we have as a given that $Dog(c)$, since we know that Charlie is a dog. Now, clearly we should not be able to infer that everything is a dog just because Charlie is a dog. And that is why we mandate the constant $a$ in the rule to be a temporary name that we use to denote "some arbitrary object from our domain ... let's call it $a$" In fact, many systems require you to explicitly introduce this constant ... it would be the formal logical equivalent to the mathematician's "consider any object $a$".
I must say that in your description off the rule this requirement is not clear. ... so if you don't understand the rule as you yourself stated, I can understand that!
Here is a formal proof example:
$\forall x P(x)$ Premise
$\forall x Q(x)$ Premise
$\qquad a$ (here is where we introduce $a$ ... so we have to make sure that $a$ is not used earlier in the proof, i.e. it is a 'new' constant. Again, this is the equivalent of saying "let's consider any arbitrary object $a$. I use the indentation to create a temporary context for the use of this $a$ ... some systems use subproofs to do this)
$\qquad P(a)$ Universal Instantiation 1 (as we saw, this works for any constant, so also for $a$)
$\qquad Q(a)$ Universal Instantiation 2
$\qquad P(a) \land Q(a)$ Conjunction 4,5
$\forall x (P(x) \land Q(x))$ Universal Generalization 6 (or: 3 through 6) (so why can we do this? Because $a$ was used as an arbitrary constant!)
Existential Generalization
'Typical' Form:
$P(a)$
$\therefore \exists x P(x)$
Explanation:
Like Universal Instantiation, Existential Generalization should really be without any restrictions: If $a$ has property $P$, then there is something that has property $P$, whether $a$ is used to denote a specific or arbitrary object.
So here I am not sure why there is this restriction stated in your description of the rule...
Existential Instantiation
'Typical' Form:
$\exists x P(x)$
$\therefore P(a)$ ... for a new constant $a$
Explanation:
OK, so in this rule we do have to treat $a$ very carefully! Think about it: you know that something has property $P$ .. but do you know what it is? No. So, what the $a$ is representing here, is "some object that has property P ... which we know exists ... but we don't know what specific object it is ... so let's call it $a$". And again, like Universal Generalization, it is best to contrast the correct use of this rule with an incorrect one: Again, suppose we use constant $c$ to denote a specific individual: Charlie. Now, suppose we know that $\exists Dog(x)$ ... can we now infer $Dog(c)$? No! Because even though we know something is a dog, we don't know whether Charlie is a dog. So, like Universal Generalization, the $a$ represents an unknown object, but this time, we do know that $a$ has property $P$. And that also means that $a$ is not a completely arbitrary object .. meaning that we can't use it for a Universal Generalization.
Example:
$\exists x P(x)$ Premise
$\forall x (P(x) \rightarrow Q(x))$
$P(a)$ Existential Elimination (OK use of rule, since $a$ is a new constant)
$P(a) \rightarrow Q(a)$ Universal Instantiation 2
$Q(a)$ Modus Ponens 3,4
$\exists x Q(x)$ Existential Generalization 5
Note that we had to do line 3 before line 4, because if we would have first instantiated the universal with $a$, then we could not have instantiated the existential with that same $a$, since the $a$ is on longer a new constant!
Best Answer
Letting the discourse domain be implicit, your formula means $\Big(\exists y\;\big(\forall x \; y > x\big)\Big);$ here we pick a particular $y$ such that whichever $x$ we consider, $y>x.$
On the other hand, in the formula $\Big(\forall x \;\big(\exists y \; y > x\big)\Big),$ whichever $x$ we consider, the $y$ that we pick (notice that $y$ here generally depends on $x$) is such that $y>x.$
Observe that we are iteratively applying $(y>x),$ in a sense going back and forth between—rather than chronologically handling—the quantifiers.
A correction: “we can pick an integer that is greater than any/every integer we could pick”. Use ‘that’ for a restrictive clause, which adds specification or narrows in on a class of objects; use ‘which’ for a nonrestrictive clause, which adds supplementary, non-essential information. While many people disregard this distinction, it does help with precise communication, and is certainly relevant when translating formal logic to English.
Your suggestions are clear; however, a slight rephrasing (e.g., “some $y$ is greater than every $x$”) renders them ambiguous due to hanging quantifiers, so do look out for that.
In this case, ‘every’ and ‘any’ interchangeably correspond to universal quantification. However, it's worth noting that sometimes, the word ‘any’ corresponds to ‘∃’ instead of ‘∀’! To wit:
$\color\red{\textbf{If any}}$ intruder enters, the alarm will go off.
$(\color\red{\boldsymbol{\exists}} x\, Ex)\color\red{\boldsymbol{\implies}} A$
$\color\red{\textbf{If any}}$ command is understood by the dog, it is a genius.
$(\color\red{\boldsymbol{\forall}} x\, Ux)\color\red{\boldsymbol{\implies }}G$
She does $\color\red{\textbf{not}}$ have $\color\red{\textbf{any}}$ disease.
She does not have some disease.
$\color\red{\boldsymbol{\lnot \exists}} x\, Dx$
$\forall x\, \lnot Dx$
For every disease, she does not have it.
She does not have every disease.
$\lnot \forall x\, Dx$
$\exists x\, \lnot Dx$
For some disease, she does not have it.
The sentences within each group above are equivalent to one another. Although some of the black sentences sound stilted (after all, natural language is quite context-dependent), they are at least unambiguous; on the other hand, the trickier red sentences indicate that using the word ‘any’ judiciously can prevent ambiguity and mistranslation. In particular: