Are there neighborhoods to avoid in Florence?
Well I'm sure there are as there are places in any city you'd be wise to avoid at 2 a.m after way to much vino. In all my trips to Florence I've never felt uneasy walking back from a late dinner etc. (and when I'm with my sister it's just us two white haired ladies). Probably because we pretty much confine myself to the historical area.
Avoid the area that is very close to the train station late at night. It is very safe during the day, but just a little "sketchy" late night. This is probably common advice in any city.