I'll help best I can. As far as safety is concerned, bad areas are Gardere, Scotlandville and various pockets near downtown like Myrtle Walk, Thomas Delpit, some areas north of Florida Blvd and near Airline north east of downtown. Watching the local news for a week can quickly give you an idea of where the violent crime centers are. Property crime is a lot more sporadic.
Some of the magnet schools have quality education. McKinley, Istrouma, BR High tend to have good reputations. There's always private, but it's generally expensive. I have no kids so I can't give real opinions but only base mine from cousins who are in school and friends who teach at some schools.
With the suburbs, you tend to get better quality schools but in the case of Livingston Parish you lose diversity and modernism. Not all, but many people who live in the northern and eastern suburbs moved there in the past to "get away from the city". Sure, it has it's walmarts and bass pro and plenty of shops and such, but the people aren't as open to change, and I find them all hypocritical because they love to proclaim how great their little neighborhood is but they ***** and whine when more people try to move in and how much traffic it creates.
The westbank is a lot slower. It's old and full of people who never wanted to move into the city. The area generally lacks in economics (and education) so the standard of living is relatively poor over there. But honestly, if I was buying a house and wanted to live outside of the city, I'd head to the westbank just because you can get more for your money and it's generally quicker and easier to get to downtown than from the eastern suburbs. And it helps that I have family on that side of the river anyway.

Depending on where you come from, you're definitely going to make less here. Statewide, quality of life is lower and the cost of living is a lot less, but in Baton Rouge it is fairly high. Do your research before settling on major expenses like a house. The need for healthcare professionals should be high but I don't know about the pay. I have one friend working in the field here, and one in L.A. Obviously the one in L.A. is making a killing, but the cost of living is so high out there. Looking at the quality of life of both people, I'd say they're right about the same.