"There are many bars downtown" is how I would say it.
I would add "in" if I was specifying which downtown I was referring to: "There are many bars in downtown Atlanta."
"City center" is extremely formal; it might be heard in a speech or statement given by a city official. One would most likely see it written in "official" documents (civic, political, etc.) or in attempts to rebrand/sell/promote downtown areas while avoiding the negative connotations some people have with the term "downtown".
I'm still learning - so no guarantees - But I would say that "det er" and "det finnes" are interchangeable if it has to do with the placement of something (it is there/one can find it there), but not if one uses "det er" in context with an adjective like for example "it is - nice/green/ tall ...").
It's in no way incorrect, however, no native American English speaker would ever say this regardless of accuracy. At least that's the case in my experience, never in my life have I ever heard anyone say "city center", it's always "downtown". I suppose it could be a regional thing, it still sounds weird though.