I've lived in Brooklyn ever since moving to NYC in the late 90's, and for much of that time I was a reverse-commuter (teaching or taking evening classes at various times) and can say it is very practical and easy and, yes, is less stressful than moving with the rush-hour flow. And, although I can't speak to what it's like to live in NJ, my commute to work when I lived in Fort Greene was a half hour door to door to get here to the Financial District. It took me less time to get here than it did those of my colleagues who live in Manhattan.
As for your question about neighborhoods...names don't really suffice. Real estate agents manipulate them pretty heavily at the fringes anyway, and to know what you're getting in Brooklyn, you pretty much have to go there and get a feel for the place. I live in Flatbush now in a really interesting, diverse, beautiful area; other parts of Flatbush are among some of the worst parts of the city. And of course it changes--so there were places I wouldn't have gone after dark 7 or 8 years ago that now are great places with affordable rents and new restaurants, bars, shops, etc. springing up all over the place.
The funny thing is that the places you list as ones to avoid--or Flatbush, Crown Heights, Bed-Stuy, at least, all three of which I know to some degree--are some of the first places I would look if I were going to be a student in NYC. Obviously you need to get a feel for the particular places, but my experience of Brooklyn is that its supposed dangers are 99.8% a pose. It helps keep prices down--and, not coincidentally, Manhattanites away.
