This is a somewhat delicate topic and I think what has been proposed so far did not capture fully what is going on. (Probably I am missing something, too, so maybe this is just a very extended comment.)

From OPs last formula identity for $\|Eu\|_{W^{s,p}(\mathbb{R}^d)}$, a short calculation shows that zero extension is in fact a bounded linear operator $$W^{s,p}(\Omega) \cap L^p(\Omega;\operatorname{dist}_{\partial\Omega}^{-s}) \to W^{s,p}(\mathbb{R}^d).$$

However, in general, the space defined in OP $$W^{s,p}_0(\Omega) := \overline{C_0^\infty(\Omega)}^{\|\cdot\|_{W^{s,p}(\Omega)}}$$ is *not* embedded into the weighted space $W^{s,p}(\Omega) \cap L^p(\Omega;\operatorname{dist}_{\partial\Omega}^{-s})$. In fact, this requires some boundary regularity. Zero extension is still a bounded linear operator for the given $W^{s,p}_0(\Omega)$ by Ayman's abstract answer. (For some reason I dismissed his argument in an earlier version of this answer/comment.)

There is a bunch of related literature about the relation of the above mentioned fractional Sobolev spaces with zero boundary conditions, referring to $(s,p)$ [or *fractional*] *Hardy inequalities*, with sufficient conditions. See for instance *On density of compactly supported smooth functions in fractional Sobolev spaces* by Dyda and Kijaczko (in particular Section 5) and the references there.

Note that it *is* classical that $W^{1,p}(\Omega) \cap L^p(\Omega;\operatorname{dist}_{\partial\Omega}^{-1})$ is embedded into $W^{1,p}_0(\Omega)$ (with the analogous definition as above) irrespective of boundary regularity. The other way around, again, requires boundary regularity. I would expect the same to be the case for the present case of fractional Sobolev spaces, but I do not know a striking reference right now.