When the domain $I$ of multi-variable function $f: I \rightarrow \mathbb{R}$ is given as $I=[a,b] \times [c,d] \times ...$, then are the cross products ordered?
I.e. $x \in [a,b]$, $y \in [c,d]$, ..
And when one integrates, then one integrates using these intervals, since changing the variables does not necessarily produce identical result.
I.e. the integral must be
$$\int_a^b \int_c^d ... \space ...dy dx$$
Yes, the cartesian product is ordered.
For example, let $I=]0,1]\times[0,1[$ and $f\colon I\rightarrow\mathbb{R}$ defined by: $$f(s,t)=\frac{1}{s(1-t)}.$$ The map $f$ is not defined if either $s=0$ or $t=1$. You cannot change the domain of $f$ to $[0,1[\times]0,1]$.
About the integral you are correct, however if $f$ is continuous on a product of segments, you can exchange the variables of integration at will. This is an easy result (no need for Fubini-Lebesgue theorem).