Is working in public hospitals dangerous for your health?

Any hospital may be. Depends on the job, exposure to patients, body fluids, etc. All hospitals have sick patients, lots of germs floating around. Any one working with "sharps"-needles, etc. Has risk of needlestick infection. Be careful, wash your hands often, or use alcohol hand sanitizers, e.g. Purell. Private hospitals have the same germs too. Worried about crime, neighborhoods, etc.? Different issues.
Possibly. The germs in hospitals are more serious and resistant to antibiotics - stay away if you can.