Yup. Food trucks in my area (LA County, but not Los Angeles itself) definitely cook in their trucks, but they effectively cluster into what I call "food truck cafeterias", side streets that seem to be set aside for food truck usage.
It's actually pretty convenient, since two streets over from my office there's usually a dozen or so trucks all available on a low-traffic street.
I’m not so certain. I know of prominently-positioned pizza trucks in SF that are clearly cooking their pizzas there. I doubt they are doing that illegally.
"All mobile food vendors must operate in conjunction with a food facility, also known as a commissary, such as a licensed commercial kitchen. Thus, you will need to rent a commercial space for preparing food, and storing your inventory."
Another commenter mentioned that there may be various classifications of food such that a Pizza truck may be able to get around it due to the nature of the ingredients but I'm unfamiliar as to where those lines are drawn
A lot of the laws differentiate between foods that are dangerous when left out (eg meat) and those that aren’t (pizza supplies). I think it’s called cottage food laws?
