I have read on many sites(per example this MSE' answer) that a $gof$ composite function is "well defined" or "is possible" when:
$f: A \to B$, $g: B \to C$, $gof:A\to C$
What this mean? is definition or what?
Why the domain of $gof$ is $A$? it is not supposed to be the intersection of $x$ in $Dom(f)$ and $f(x)$ in $Dom(g)$? The only reason I see for the definition of $gof$ is that enters $A$ values(because is the domain of the first function $f$) and get $C$ possible values(because is the codomain of the last function)
Why the domain of $g$ is $B$? i think that the domain of $g$ in a composition will be $B$ if the sets $Range(f)$ and $Dom(g)$ are equal, but $Range(f)$ can be subset of $Dom(g)$
You're forgetting that, in the composition, $f$ is applied first. Thus, $f$ takes all of the elements of $A$ and sends them to a subset of $B$ called $\text{range}(f)$, and then $g$ (since the composition is $g(f(x))$) sends all of the elements of $\text{range}(f) \subseteq B$ to a subset of $C$.
Individually $f,g$ are separate functions, so $g$ could have a domain bigger than $\text{range}(f)$ if desired, since $g \circ f$ only is sampling from a subset of the domain of $g$ (the subset $f$ maps to) and not the entire domain.