Let $G$ be a $\mathbb{P}$-generic filter over a ground model $M$ of ZFC and $G\notin M$. Are all objects built by this generic filter necessarily out of the ground model $M$? Particularly is limit of this generic filter $\cap G$ out of $M$? (i.e. $\cup G\notin M$)?
My question is "does $G\notin M$ imply $\cup G\notin M$?" we know both of $G, \cup G$ are in $M[G]$ of course.
It's not always true that $\bigcup G\notin M$.
For example, Sacks forcing with perfect trees, $\bigcup G$ is just the full binary tree, since the order is inclusion (stronger is smaller).
Every forcing which can be described with the relation "$p$ is stronger than $q$" if $p\subseteq q$ will invariably have that $\bigcup G=\max\Bbb P\in M$. These include every forcing where we force with a complete Boolean algebra, which in turn can be assumed to be every forcing (of course sometimes we prefer to present a partial order which is not a complete Boolean algebra and work with it instead).
Usually, however, we are less interested in the generic filter, but we are interested in some object canonical described from the sets within the generic filter somehow. In the simple cases, this would be $\bigcup G$, or sometimes $\bigcap G$. In other cases it will be something more complicated.
What we often want to achieve is some "relatively canonical" object $x_G$ such that $G$ can be reconstructed from $\Bbb P$ and $x_G$ somehow. How we generate $x_G$ depends very much on our forcing and what we are trying to do.