Do both ships have to be stationary [during drone operations]? Can a Vulcan refuel you while both ships are moving, since the drones can fly around?

It can only help. So the drones will be flyable things, so they will have their own speed limits. So I wouldn't expect people to be afterburning whilst trying to have this little drone catch up to fuel them. So you'll be able to do it in motion, just at a considerate speed to each other. For the other actions, there's nothing stopping you launching your drones and the Vulcan just staying where it is and the drones go off. That ship could be flying around still, but it's going to get harder the more that person starts moving around.