On the weak closedness I have some difficulties in this question.
Let $X$ be a nonreflexive Banach space and $K\subset X$ be a nonempty, convex, bounded and closed in norm. We consider $K$ as a subset of $X^{**}$. I would like to ask whether $K$ is closed w.r.t the topology $\sigma(X^{**}, X^*)$.
Thank you for all comments and kind help.   

No, this fails in every nonreflexive space for $K=\{x\in X:\|x\|\le 1\}$, the closed unit ball of $X$. 
Indeed, any neighborhood of a point $p\in X^{**}$ contains the intersection of finitely many "slabs" $\{x^{**}\in X^{**}: a_j< \langle x^{**}, x^*_j\rangle < b_j \}$ for some $x^*_j\in X^*$and $a_j,b_j\in \mathbb R$. It is easy to see that $\bigcap_j \{x\in X: a_j< \langle x, x^*_j\rangle < b_j \}$ is nonempty as well; indeed,  $\bigcap_j \{x \in X: \langle x, x^*_j\rangle = (a_j+b_j)/2\}$ is a subspace of finite codimension. Thus, $X$ is dense in $X^{**}$. Since the embedding of $X$ into $X^{**}$ is isometric, the unit ball of $X$ is dense in the unit ball of $X^{**}$.
