They're just resting their eyelids!
I should hope so...
Why can't I say "piloterna är inte sover" ?
There's no such construction in Swedish. Both are sleeping and sleep correspond to sover.
Why is 'piloterna är sover inte' incorrect?
Swedish doesn't have the am/are/is verb + ing construct.
If I want to say "the pilots do not sleep", how would that be? Is there a way to make the difference clear?
That would be the same :). You need a work-around to tell the difference, like "right now", "normally" etc.