How could you run getting a hospital ship to a plague moon before everyone dies?
Well, you're going to need maneuvering power/acceleration. In a time sensitive disaster response situation, "the faster the better" ... BUT ... you're also going to need to "bring enough STUFF™" with you to make the trip worth the effort. No point arriving first if all you brought with you was a thimbleful of water to a population dying of thirst (for example). As with everything in engineering, it's the tradeoffs that become important since you need to find a "sweet spot" balance point between a pile of competing priorities. The trick is figuring out which objectives need to be prioritized (and in what order) to engineer a capability that meets the need.
What rolls are needed? And how often? I understand how getting more speed from the maneuver drive works Engineering-wise, but what does the Pilot and Astrogator need to do?
Think of the Pilot as being the "local controller" of the craft. The Pilot is the one who
flies the course.
The Navigator is the "long distance plotter" for the craft. The Navigator is the one who
plans and determines the course which the Pilot is then obliged to fly.
If you want an analogy to ground car track racing (so as to have something easier to grab a hold of conceptually), there is a notion in track racing that there exists a LINE through the track which is the optimal path to complete the track in the least amount of time. It is then the responsibility of the drivers to stay as close to that LINE as possible, since deviating away from that LINE will negatively impact their time score and position within the race. The more time a driver spends "riding the LINE" as closely as possible, the better/faster/more efficient their race performance is.
Pulling this back to maneuvering spacecraft, the Navigator plots the LINE ... and the Pilot then attempts to fly along it as closely/cleanly as possible.
Under nominal circumstances, a Navigator is only really needed for acceleration voyages lasting 12+ hours. Anything that takes 12- hours to transit will not require the specific skillset of a Navigator and can be done by a Pilot alone (the distances are short enough that "everything is local" between origin and destination). If you're needing to make an interplanetary transit (which will typically take longer than 12 hours), best practice would be to have a qualified Navigator to generate the course plot which the Pilot then flies.
Has anyone run a race in a game?
... Maybe ...? 
