From a Wolverhampton perspective, it's not that bad a place I don't think generally. I know it has a certain reputation, but there are certainly places that are worse. Good for younger families, cheap, good transport links. Just not like some of the more 'niace' places, but friendly people.
In terms of areas, generally the further away from the centre, the better! If you look at a map of Wolverhampton and draw a line down the middle through the ring road, then anything to the West of that is a better area on the whole. As others have mentioned areas like Compton and Tettenhall are good, but probably don't fit your criteria so much. Likewise, surrounding towns/villages like Albrighton/Shifnal are popular and have good links to B/ham, but are pretty white, and probably won't have the Church/community aspect you're looking for. I'd suggest Penn/Finchfield/Merry Hill (not to be confused with the shopping centre of the same name...) would fit better as more diverse, would be able to get a nice house for that money there, and are more ranges of Churches and associated community.
Oh, and yes, the City Centre (more town except in name) has been on the downhill slide for a while. Quite a lot of empty shops, and increasing numbers of homeless. Several other shopping options available though.