Q1) The function $\boldsymbol 1_{\mathbb Q\cap [-2,2]}$ is in $W^{1,1}(-2,2)$ whereas it's no where continuous (and thus nowhere differentiable), whereas $\boldsymbol 1_{[0,1]}$ is not in $W^{1,1}(-2,2)$ whereas it's differentiable everywhere except at two points that are $0$ and $1$... Isn't it strange ?
Q2) Let $f\in \mathcal C^1([a,b]\times \mathbb R\times \mathbb R)$ and $u\in W^{1,\infty }(a,b)$. I have the following equation :
$$\int_a^b [f_u(x,u(x),u'(x))-\frac{d}{dx}f_\xi(x,u(x),u'(x))]v=0$$ for all $v\in \mathcal C_0^\infty ([a,b])$. By fundamental lemma of calculus of variation, the lead to $$f_u(x,u(x),u'(x))=\frac{d}{dx}f_\xi(x,u(x),u'(x)),\quad a.e.$$
Now it is said that since $f_u(x,u(x),u'(x))$ is continuous, the equation hold for every $x\in (a,b)$. But this is wrong argument, no ? Indeed, we have for example $\boldsymbol 1_{\mathbb Q }=0$ a.e., $0$ is continuous, but tu equation doesn't hold every where...
1) It's because the function $\boldsymbol 1_{\mathbb Q\cap [-2,2]}$ has a continuous class representant (I'm not sure about this name, if $[x]$ denote the class of $x$, I denote $x$ the class representant, but it has probably an other name in english) whereas $\boldsymbol 1_{[0,1]}$ has not such a representant.
2) To my opinion, it miss something (as $\varphi\in W^{2,p}$) or something like that. Are you sure about all hypothesis ?