It seems like both the class and the color are determined by your location (can vary from one camp to another on the same planet) and the Head you use.
I just upgraded it at the space station to be S-Class
Yes, S-Class Staffs are obtainable in a similar way to multitools. With Staffs, the Head piece determines class and (for the most part) color. So, if possible, check every different head at every different Autophage Terminal in order to find S-Class Staffs. It should be considerably faster than hunting Multitools.
I haven't as much as I'd like yet, but from what I've seen so far it's the same as a multi tool. Lasers and fire coming from the end of a staff does look cool!