They're bordering places, there is no real distinction between the West of Birmingham and all of the Black Country, there is no equivalent to the Meridian Gap
Most pubs around the ground are pretty strictly home fans only. Last time I was up there I just went for a drink in the city centre, it's walkable and there are some decent pubs. Devonshire Cat is good but I had heard it had closed down.
The black country has no formal definition and I've heard previously that the entire city of Wolverhampton is not in it.
Thankfully I'm not a brummie like you so can plead ignorance