If $u_{n}\rightharpoonup u$ in $W_{0}^{1,p}\left(\Omega\right)$ , do we have $u_{n}^{+}\rightharpoonup u^{+}$? If $u_{n}\rightharpoonup u$
  in $W_{0}^{1,p}\left(\Omega\right)$
 , do we have $u_{n}^{+}\rightharpoonup u^{+}$
  and $u_{n}^{-}\rightharpoonup u^{-}$
  in $W_{0}^{1,p}\left(\Omega\right)$
  and vise versa?

The question first is whether $u_n^+$ is in $W_0^{1,p}$ for all $n$ if $u_n\in W_0^{1,p}$. Since $u\in W_0^{1,p}$, there exists a sequence $\phi_m^n\in C_0^\infty$ converges to $u_n$ in $W_0^{1,p}$. Then $\phi_m^{n,+}$ converges to $u_n^+$ in $L^p$. It is also not difficult to prove that $\phi_m^{n,+}$ is Lipschitz continuous, then it has a.e. derivative. For $x$ with $\phi_m^{n}(x)>0$, we clearly have $\partial_i\phi_m^{n,+}(x)=\partial_i\phi_m^{n}(x)$; for For $x$ with $\phi_m^{n}(x)<0$, we clearly have $\partial_i\phi_m^{n,+}(x)=0$; for $x$ with $\phi_m^{n}(x)=0$, if its every small neighborhood contains $x$ with $\phi_m^{n}=0$, then it has at most derivative $0$; if its every small neighborhood contains $x$ with $\phi_m^{n}\neq 0$, it has at most derivative $\partial_i\phi_m^{n}(x)$. This means that $\partial_i\phi_m^{n,+}$ is in fact bounded by $\partial_i\phi_m^n$ in $L^p$, and since $\partial_i\phi_m^n$ converges to $\partial_i u_n$, we know $\partial_i\phi_m^{n,+}$ is bounded in $L^p$ (for $1< p<\infty$, $L^p$ is reflexive), then it has a weakly convergent subsequence which the limit is denoted by $f$. Then
\begin{align}
\int u_n^{+}\partial_i\psi&=\lim_{m\rightarrow\infty}\int\phi_m^{n,+}\partial_i\psi\\
&=-\lim_{m\rightarrow\infty}\int\partial_i\phi_m^{n,+}\psi\\
&=-\int f\psi.
\end{align}
Therefore $u_n^+\in W_0^{1,p}$.
Suppose $1< p<\infty$ and $u\in W_0^{1,p}$. Since $u_n$ is weakly convergent, $u_n$ is bounded, then $u_n^+$ is also bounded in $W_0^{1,p}$. Let $u_{n_j}^+$ be a subsequence of $u_n^+$ (we omit $j$ in the following). For $1< p<\infty$, $W_0^{1,p}$ is reflexive. Then $u_n^+$ has a weakly convergent subsequence. Say the limit is $u^1$. Now we know from the Sobolev embedding theorem that the embbeding from $W_0^1{p}$ to $L^p$ is compact. So the sequence $u_n$ and $u_n^+$ are strong convergent in $L^p$. $L^p$ convergence also implies almost sure convergence for subsequence. One must then have $u^+=u^1$ almost sure in $\Omega$. Therefore every subsequence of $u_n^+$ has a weakly convergent subsequence with limit $u^+$. This proves the statement. Vice versa is trivial and even for $1\leq p\leq\infty$.

Edit 1: to be able to apply Sobolev's embeddings lemma we also have to give some domain conditions. A sufficient condition is that the domain is a lipschitz domain.
Edit 2: I have made a big mistake before (didn't consider the case for $\phi_m^{n,+}<0$) and now it is fixed.
