What are all underlying assumptions in the definition of Cartesian Product in Naive Set Theory?

The way you should read $\{x : P(x)\}$ is "The set of all $x$ that satisfy the condition $P(x)$." To define $A \times B$ this way, we ask ourselves "What is the condition that a particular element $x$ must satisfy to be in $A \times B$?"

The answer to that question is "It must be of the form $(a,b)$ for some $a \in A$ and for some $b\in B$." So that's what we write in the set definition.

You want to have a "for all" there of the related statement "For all $a \in A$ and for all $b \in B$, $(a,b)$ is an element of $A \times B$." But that's not the kind of statement we put in a set definition. If you were to write $$ \{ x: x = (a,b) \text{ for all $a \in A$ and $b\in B$}\} $$ then it would mean something different. It would mean that to test if an element $x$ is in this set, we would construct ordered pairs $(a,b)$ for all $a\in A$ and $b \in B$, and test if $x$ is equal to all of them simultaneously. This is impossible unless we're in the exceptional case $|A|=|B|=1$, where there is only one such pair.

(Or unless $|A|=0$ or $|B|=0$, in which case this definition runs into weird "naive set theory doesn't always work well" issues.)

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For the second question, the phrase "the fact that" can and should be omitted from all sentences in which it occurs. So you can read the quote as

The set $A \times B$ is characterized by $$ A \times B = \{ x: x = (a,b) \text{ for some $a \in A$ and for some $b\in B$}\} $$

The phrase "is characterized by" means "is the only thing satisfying the description" or "are the only things satisfying the description" depending on context.

(1) The set (for $A,B\ne\emptyset$) $$ \{ x : x = (a, b) \text{ for all } a\in A \text{ and for all } b \in B\} $$ will be empty with one exception (what exception?) because $$(a_1,b_1) = (a_2,b_2)\iff a_1 = a_2\land b_1 = b_2$$ (2) The phrase "it is characterized by the fact" means "is defined by".