Heh...thanks. I'm a big Godfather fan as you can tell.
Obviously some parts of town are more diverse than others. The surrounding counties tend to be more "white", but Davidson County (where Nashville proper is) is pretty diverse all around. Like any big city, there are predominantly African-American neighborhoods. Some are in fairly rough economic areas. But I think it's very fair to say that you could live pretty much anywhere in Davidson County and be in a racially diverse area. That's not to say there aren't diverse areas outside of Davidson, but I'd start in Davidson Co first.
As far as neighborhoods go, you can find a lot if diverse opinion on this matter all over this forum. I'd start looking in neighborhoods with a large amount of under 30 folks...Bellevue, East Nashville, West End/Vanderbilt to start. Bellevue is pretty affordable, but further out from downtown. East Nashville is a trendy place to live right now...a transition community which means there are still some rough patches. West End/Vanderbilt area is a great place, pretty young, but pretty pricey from what I gather.
I can't speak on how the ladies are these days, I've been off the market for awhile.

But I think church is a very popular place to find singles. I just don't know...luckily I haven't been on that scene for a few years.
Good luck to you and let us know if you have any more questions...and definitely search the forums. You'll find lots of good advice.