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It only counts sectors that are not regular / offgrid / blocked / home. So I'm assuming that means something that doesn't have a yellow dot in it at all per the game. I don't think sectors can "become" offgrid - they are either offgrid (i.e. spawn with asteroids or something else in them) or not.
The one exception I made was that the encounters will not spawn in a sector with any stations. I didn't want the player switching back to their economy sector that they built in the core and having a boss spawn in there and start wrecking the place. I'm sure that would be a way to get some rage comments. However, this does NOT reset the counter.
There is a 30 minute cooldown that they share with swoks / the ai / etc. that starts after the boss is killed, not after the boss is encounterd.
They also have their own 180 minute cooldown.
I think that should answer all of your questions.