The Sobolev seminorm on $H^s$ is $$|f| = \int_{\Omega}\int_{\Omega} \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2s-1}}dxdy$$ My question is, is this integral a double integral or a iterated integral?
i.e. can I integrate one variable first and then the other, so can I write $$\int_{\Omega}\int_{\Omega} \frac{f(x)g(y)}{|x-y|^{n+2s-1}}dxdy=\int_{\Omega}\bigg(\int_{\Omega} \frac{f(x)}{|x-y|^{n+2s-1}}dx\bigg)g(y)dy$$ and then use a bound on the inner integral and a fact that $g \in L^1$ to show that $\tilde{f}\tilde{g}$ has finite $H^s$ seminorm, where $|\tilde f \tilde g|_{H^s}$ is less than the left hand side of the above equation.
Yes you can integrate (the first integral) with respect to $x$ and $y$ in any order you want. This is an implication of Fubini's Theorem.
Unfortunately, what you suggest to do next is not a good idea, as the difference ratio of the first integral has much better chances to be integrable, than if you split it expanding $|f(x)-g(x)|^2$.