So does this mean that I can say that, for example, $\gamma \frac{\partial u}{\partial \nu}$ has a unique continuous extension as an operator from $W^s_p(\Omega)$ onto $W^{s-1-{\frac 1 p}}_p(\Gamma)$, and so on? I.e. I don't need to worry about some product space.
Yes, the continuity of a map $(f_0,\dots,f_l)$ from some space $\Omega$ into a product space $ X_0\times \dots\times X_l$ is equivalent to the continuity of every map $f_j:\Omega\to X_j$ for $j=0,\dots,l$. This holds for topological spaces, and in particular for normed linear spaces.