I'm a Durham fresher and I am at Hatfield College (known to be notoriously posh), to simply answer your question, yes quite a few people you will meet are posh and well-off. But that does not meet they are stuck-up (sure there are a few stuck-up people, but that's everywhere). No one really discriminates based on poshness, and you will find your group of friends.
There are loads of independent shops and cafes and they are very good (highly suggest flat white and Whitechurch). The town is very beautiful and the locals are friendly as well (unless you act like a knob). I will say that there is a lot of walking, most people don't use buses, but I reckon you'll get used to this (Pro tip: choose a bailey college but trust me whatever college you'll end up in the chances are you'll really like it). The nightlife I would say is mixed, there are a few good clubs and a few were small and cramped ones (My favs are Bohemia, Babylon, and ofc Spoons)