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.

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, and I also did not fully answer OP's question, 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. It is unclear to me whether zero extension is still a bounded linear operator on $W^{s,p}_0(\Omega)$ if that space does not embed into the weighted space.

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.

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, also I cannot fully answer OPs answer, 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. Whether zero extension is still a bounded linear operator for the given $W^{s,p}_0(\Omega)$ is unclear to me and I suspect there might be a divergence here between integer and fractional Sobolev spaces due to the nonlocal nature of the Gagliardo (semi)norm.

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.

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.