Matt Wilson wrote:
Someplace in the outer zone, on some icy frozen moon. For an M dwarf flare star, how far out is safe enough?
It depends. Some flare stars get to be about as energetic as our Sun is normally when they flare; others can be considerably brighter. On a quick search, it looks as if the shielding needed for long-term protection from galactic cosmic rays (GCR) would probably be sufficient for all but the most extreme cases.
The dark side of a tide-locked world.
This would definitely be sufficient, as well as cutting your GCR dose in half.
A station orbiting in the L-2 point behind a decent-sized world, such that the world was big enough to entirely occlude the star
That last one I'm definitely curious about. Would something at Mercury's L-2 point get any sunlight?
This would work in theory, but would be hard to implement in practice. The three colinear Lagrange points (L1, L2, and L3) are inherently unstable: any perturbation tends to move an object away from them. In the case of a close-in planet (e.g., Mercury) there are relativistic forces that act as perturbations, making it effectively impossible to passively remain at the L2 point. The options are (1) constant thrust, which may not be a problem if your setting has reactionless or high-Isp thrusters, or (2) one of a class of periodic orbits around the L2 point. This latter option may not be very helpful, however, as the orbits tend to be wide (tens of thousands of kilometers). They might not fit in the umbral zone of a close-in planet unless it was Jupiter-sized.