The only hospitals I would avoid are Jackson Park and Roseland but I'm not even sure they have NICUs. With 3 years experience it depends on the hospital but I imagine you'd be making $32-$35. If you do agency you can make around $45 - $52.
I'd recommend finding what area you think you'd like to live in and then looking at the job opportunities from there on out. Figure out if you can walk, need to commute, or if you have a car can drive.
And as for a car it's really up to you. I've lived in the suburbs most of my life and just don't feel comfortable not having one. I could also take the bus to my job, get a Lyft/Uber, or walk for 35 minutes if I wanted (I don't, and probably never will lol).
So again the car is up to you, but make sure wherever you live has most of what you need in walking distance (grocery store, public transportation, pharmacy, etc). Most places you want to go you can use public transportation. But also keep in mind public transportation doesn't all run 24/7 here and sometimes certain routes are only on certain days so do your research in that area too.
Hope the move goes smoothly