Consider the function $$ f(x,y)=h(x)\times g(y) $$ By Fubini's Theorem $$ \int_{X}h(x)dx \times \int_{Y}g(y)dy= \int_{X\times Y}h(x)g(y)d(x,y) $$
Questions:
(1) Is this correct?
(2) Can be extended to the product of any number of functions?
(3) Are there particular conditions to check other than measurability?
Yes, you are correct that the product of the integrals equals the integral of the products. And yes, this can be extended to triple products, and quadruple products, and so on.
In general, Fubini's theorem only holds when $f(x,y)$ is integrable, i.e. when $\int_{X \times Y} |f(x,y)| d(x,y) < \infty.$ But in your situation, $f(x,y)$ is a product of $h(x)$ and $g(y)$, which are integrable by assumption. Hence the product $f(x,y)=h(x)g(y)$ is guaranteed to be integrable and everything works okay.