I imagine multi-stage missiles must have some purpose, but I haven't yet figured out when I might want to use them.
Mines, efficient long range launch stage with speedy short range munition, MIRVS.
I've also gotten the idea that you can create 'buoys' which are missiles that carry a payload of a sensor, rather than a warhead, and they last a long time? Or something?
Yes. They last as long as they are not shot at by anti-missile fire.
I have this dim memory of long-ago creating missiles that could track targets under their own power, and therefore operate far outside the targeting range of the ships that fired them. (I may be remembering a different game altogether.)
You just need an active sensor (on the missile) and a location. If the missile reaches the destination without seeing a target it will wait there until it does, or detonate if fuel runs out.