I think the El closes from midnight to 5 am or something. You can look it up online... there should be info and maps. The suburbs can be a little cheaper, but not much. Also, a short distance from the hospital can be a long commute. As a MS4 I lived 3 miles from one hospital, and it routinely took 30-45 min to drive each way. No joke. My PD said I could run to and from work faster, and he was right.
To get to the west suburbs fom Cook Co, you would take I290 and it is notoriously terrible for traffic at any time of day or night. Plus, Cook is only about 2 miles from "the merge", or the junction of the Dan Ryan (I94 south of the city), Kennedy (I94 north of the city) and I290, which is a nightmare of slowness. Thus any back up eastbound will translate to a long back up on I290.
You may be a better driver than I am, but since I have been in two accidents post call (one into a building), I am glad I live within a 5 minute drive of my hospital. I would get in touch with current residents and see where they live.
Lastly, "safe" on public transportation is partly a matter of awareness. Any part of the city can be dangerous if you are not aware of your surroundings and your neighborhood. People in the city mind their own business, and are not chatty. When I moved back South, people thought I was rude, because I don't strike up conversations on the elevator, at the gas pump, in lines, etc.
I never felt unsafe, but I knew the few areas to avoid... the area around Cook Co. is safe if you are aware and careful about going out alone at night. There are some very unsafe areas within blocks of the hospital, but again, I did several rotations as a student, and did not worry a lot. Talk to your coworkers; they will be a good source of info.
And to reiterate, I love the city; Chicago is a great place to live. Too bad about the malpractice issues.