If $u$ and $w$ belongs to the same connected components, does there exist any $u-w$ path containing $v$? 
If $u$ and $w$ belongs to the same connected components, does there exist any $u-w$ path containing $v$?

Not in $G_1 \cup G_2$, since $v$ is contained neither in $G_1$ nor in $G_2$. Indeed, $G_1 \cup G_2 = G \setminus \{v\}$. However, there exists a path in $G$ because $G$ is connected.
