IF all computers are on the exact same build of PrPro. Understand of course that if everything of the project is on one flashdrive, performance in playback & etc. with much in the way of effects or tons of cuts is going to be ... s l o w . But perhaps doable on say a school project, which probably doesn't have the total assets of say a season-length tv sitcom.
Theoretically, yes. However the speed of a USB flash drive is probably not quick enough to make your editing experience very good. I would recommend instead you use an external hard drive, preferably with a fast interface like USB 3 to shuttle your project between home and school. You may want to point your scratch disks to the local drives.