I never found the winters to be a problem. I moved from Maryland and I did have to learn to wear a hat, gloves and scarf though. Gloves are not optional when walking home with groceries in 10 degree weather -- it will seriously hurt if you don't wear them. I didn't understand scarfs until I moved here. You'll find you'll be perfectly comfortable if you dress the part -- you just can't be one of those people who's too cool for a hat and expect to like our winters. Get a good coat and layer. And I discovered overshoes a couple winters ago, which I now swear by when there's a big snow --
NEOS® Performance Overshoes - NEOS -- totally worth it not to have to worry about getting your feet all wet in the freezing cold.
If you're concerned about diversity, you might check out Pilsen or Uptown, which are both very walkable but not as "gentrified." Also Ukrainian Village south of Wicker Park has the benefit of being a short walk away without actually being Wicker Park. I don't know where the grocery store in Pilsen is though. Uptown is close to the lake, which is nice, and it has lots of incredible old buildings. Of course, if you move there, you may be contributing to gentrification.
Otherwise, you might like Gold Coast or Lakeview if Wicker Park is too hipstery and Lincoln Park too yuppie for you. Gold Coast is very dense, close to downtown, I think feels about as close to Manhattan as Chicago gets. Lots of street life, has lots of rich people, but also lots of poor. Seems to have a lot of Europeans.
I've lived in Lakeview for over four years without a car. I like it because it's an odd mix of mini-neighborhoods that are unlikely neighbors (gay neighborhood next to the jock neighborhood next to the punk neighborhood next to the young family neighborhood), it's really well connected via transit, close to the lake and other neighborhoods, and it has all of your everyday needs, plus lots of cheap but good places to eat. The Belmont stop serves three train lines. It's a more stable neighborhood -- hasn't gone through the rapid changes and shifts some other neighborhoods have seen in recent years, so it's retained some of the more mundane businesses you might otherwise not know you'd miss, like hardware stores, locksmiths, lighting stores, dentists, veterinarians, etc.
Also of note, if you live along the lake on the north side, it's pretty much a nice walk to any other neighborhood along the lake on the north side. Neighborhoods west may feel a little more isolated without a car, unless you bike a lot.