Is Neighborhood an open set?

There are different notions of neighbourhoods floating around. Usually one calls Rudin's approach open neighbourhoods to avoid confusion. Whereas the one you just cited is just the ordinary neighbourhood definition (if people use open neighbourhoods instead of neighbourhoods always, they usually say that in the introduction). Generally speaking a neighbourhood of $x$ is just a set $X$ such that it contains an open set $U$ with $x \in U$.

In particular every neighbourhood contains an open neighbourhood, and so passing from general neighbourhoods to open ones contained in them is not too hard. Passing to closed neighbourhoods in a given neighbourhood however is more difficult and is one of the reasons one likes to have a regular Hausdorff (also known as $T_3$) spaces.

Clearly not. Take $X = \mathbb R$ with the usual metric, then $(-1, 1]$ contains a ball $(-1/2, 1/2)$ centered at $0$, so $(-1, 1]$ is a neighborhood of $0$, but itself is not open.