Some thoughts from a native Chicagoan (grew up in Highland Park, which is directly north of Northbrook, and lived for years in Lakeview):
You can live in Chicago without a car. I did for almost 10 years. If you live near the Red or Blue lines, they run 24/7, as do many of the main bus routes. In my experience, owning a car in Chicago is expensive and frustrating. However, most transplants to Chicago own cars - as far as I can tell, because they grew up with cars in some suburb somewhere and are bizarrely incapable of giving up the crutch. If you really need to buy three tons of toilet paper at one go, Zipcar is very easy and far more economical than paying for gas, insurance, etc.
Lakeview is a decent neighborhood, although it's become much blander than it used to be, as has much of the North Side. It's now pretty much indistinguishable from Lincoln Park and Bucktown - home to frat bros and woo girls from Big 10 and Midwest state schools, who spend their time shopping at big-box stores, getting plastered at sports bars, and eating at mediocre Mexican restaurants with margarita specials. Even the gay bars (Lakeview was once known as Boystown before the Bro Invasion of the late 90s) are generally lame.
Other (and cheaper) neighborhood options would be Ravenswood or Roscoe Village, which are both a bit west of Lakeview and haven't totally turned frat yet. Edgewater (which also includes Andersonville) is pretty cool as well and more convenient to north suburban commutes. All these neighborhoods still have their local character intact to varying degrees, with a nice mix of local businesses and places to hang out, but without attracting every douchebro in the Midwest. Edgewater is also right by the lake, with great beach and park access for biking, running, volleyball, etc. You can probably find a very nice two-bedroom for $1200 or less in these areas.
As has already been mentioned upthread, Evanston is a pretty cool city as well, although in my opinion if you're not a Northwestern student or a suburban empty nester, you may feel out of place. Also, it's more expensive than the neighborhoods I mentioned above for rent.
The West Loop/Near West Side to me feels very synthetic. Lots of new condo buildings and flash restaurants, but no feeling of a real neighborhood. It was very industrial until quite recently, and while convenient to the Loop feels very much like a soulless place. It's got kind of a rich dbag vibe to be honest - lots of 20/30-something bros working in finance, "day trading" etc. Also, I don't think there's much of a rental market in that area, except perhaps for people renting out luxury condos.
Crime in Chicago is, as you noted, highly concentrated in terms of both geography and demographic. In fact, the murder rate was much higher in the 1990s. I think the reasons Chicago's crime problem gets so much attention right now is that NY's rate has plunged, so Chicago's headline numbers look so much bigger. Exercise common sense, and you're likely to be fine.
The city-to-Northbrook commute sounds like a fate worse than death, but then again, so does living in Phoenix, and you seem to have survived. I know people who have done commutes like that using Metra and a shuttle, and they seem to do OK. That would be the one thing in your scenario that gives me pause, though.