This has to do with a very annoying NPC in the game:

Is it mandatory to go hunting Gray Eye? I've done it a few times, but eventually on replays I simply cancel the mission. I see no reason to slow myself down, especially when flying through pirate- or Korath infested space, for a personal vendetta on Moby Dick's space cousin that holds no value for me, except for the skirmish experience I get while waiting for the old codger to refuel in systems I usually would have jumped through.

Is there any reason to go ahead and hunt Gray Eye (I've once found him IN REMNANT SPACE! Programmers, can you maybe do something about that, please?), or is it fine to get roped in and cancel it immediately?

Gray Eye is from the Lost in Midnight plugin, not the base game. Apparently in newer versions of the plugin the places the NPC can be found in has been fixed. For your other questions, you'll probably need to reach out to MidnightPlugins on Github or Discord.