Bicycling in Palermo

Perhaps there are parts of town where biking would be OK; I think I remember some pedestrian streets--maybe bikes are OK there. In general, I'd say I was nervous enough just getting across the street on foot and would never want to attempt any other mode of transportation in the city.

No, nor would I.
Palermo in terms of traffic is one of the most chaotic cities that I've ever had to navigate.
If you are thinking of using a bike for local transport, then I really wouldn't recommend it.
If you are a keen and fit cyclist looking to explore parts of Sicily, then I might look at the areas around Cefalú up into the Madonie park. It is lovely scenically, but very steep.

we brought our fold up bicycles with us to Palermo. yes the traffic is crazy, but you don’t have to ride on the main roads. we saw many people on bicycles,...but depending on where you are staying, where you want to go, the time of day, etc, riding a bike can be lots of fun! there are bicycle tours listed on airbnb palermo under “experiences”, check it out!