Question is exactly as the title says, is neighborhood data included in the database extract? Specifically for the US, I don't see a mention of neighborhood in the readme, but is this information in the admin3 or admin4 code?
neighbourhoods normally don't have admin functions and are classified with featurecode PPLX.
when they do have admin functions then there are often two entries one for the admin feature (ADM3 or lower) and an additional PPLX feature if the feeling of 'neighbourhood' is important together with the admin function.
the relation of a post to a neighbourhood can be additionally modeled in the hierachy table, this is different tot he admin relation in the adminX codes.