An equivalent norm on $W_0^{1,2}(\Omega)$

In some online notes regarding the equivalence of norms on Sobolev spaces, I came across the following point:

"In the case $$W_0^{1,2}(\Omega)$$, one does not need the regularity of the boundary."

I checked Evans to see if I could gain some more insight into this point, and found the following on page 280:

"In view of this estimate, on $$W_0^{1,2}(\Omega)$$ the norm $$\|Du\|_{L_2(\Omega)}$$ is equivalent to $$\|u\|_{W^{1,2}(\Omega)}$$, if $$\Omega$$ is bounded"

where the estimate in question is Poincare's inequality; $$D$$ here denotes the gradient operator. I also consider $$\Omega$$ to be open as well as bounded in $$\mathbb R^n$$.

My problems: (i) I don't understand the part from the first quote which reads "we do not need the regularity of the boundary", and (ii), how this relates to the fact that, on $$W_0^{1,2}(\Omega)$$ one doesn't need to consider the contribution from $$\|u\|_{L_2(\Omega)}$$ in the $$W^{1,2}(\Omega)$$-norm, whereas one does in $$W_0^{1,2}(\Omega)$$. How exactly does this simplification occur? Also (iii), can the norm $$\|Du\|_{L_2(\Omega)}$$ ever be regarded as equivalent to $$\|u\|_{W^{1,2}(\Omega)}$$ on $$W^{1,2}(\Omega)$$, or is it only for $$W_0^{1,2}(\Omega)$$?

(1) $$W^{1,2}_0$$ is defined as the completion of the space of smooth, compactly supported functions. So, every function in $$W^{1,2}_0$$ can be approximated by a smooth function. This is not true in general for $$W^{1,2}$$, where one needs regularity of the boundary of $$\Omega$$ to get this property. Approximation by smooth functions is essential, as embedding theorems etc are first proved for smooth functions.
(2) $$W^{1,2}_0$$ with $$\|\cdot\|_{W^{1,2}}$$ is a Banach space by construction. By Poincare inequality, the norm $$u\mapsto \|Du\|_{L^2}$$ is equivalent.
(3) This is of course not true for $$W^{1,2}$$: If $$u$$ is a non-zero constant function, then $$Du=0$$. And $$u\mapsto \|Du\|_{L^2}$$ is not a norm. This problem does not occur in $$W^{1,2}_0$$, as the only constant function in $$W^{1,2}_0$$ is the zero function.