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daf bit: Nazir 49
If a nazarite becomes ritually impure, he must shave, wait to become
ritually pure, and start over. What are the things that can make
him ritually impure? The mishna says contact with a corpse or
parts (or derivatives) thereof. What is contact? Touching, being
under the same roof as, or being above or below a corpse (without an
intervening roof). (49b)
This list has a lot in common with contact forbidden to kohanim. I don't know if that's just a similarity or if there is some causal link; I could imagine seeing priests as being akin to nazarites in that they are dedicated to ritual service, though most are not actual nazarites. (The bit about being above a body without a roof intervening is why some kohanim will not fly over graveyards. I have no idea if floors on multi-story buildings count as roofs; does a kohein have to worry about the apartment below his?)

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However, the other part of niddah is the prohibition of having relations with the niddah. This is a different effect with the same cause (Niddah implies both this prohibition and the state of tumah. Everything we do today re: niddah is a result of the prohibition and not the tumah). Then, to prevent any possibility of having relations, a variety of other things become prohibited.
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* Clearly not, since a plane is a vehicle, not a building, yes?
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As for your last question: according to this site (which seems to be reputable): "An object that is flying through the air, such as a bird or a Talis that is carried or caught up by the wind, is not an Ohel and is not Mevi or Chotzetz, even if it is one Tefach wide." So that sounds like a "no" for the plane.
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Tumah 101
Thus Kohanim today who observe this mitzva should avoid being in a situation which will make them tamei. The tumah of a Jewish corpse is transmitted in 3 ways: 1) touching 2) carrying 3) tent, which has 3 subcategories 3a) being together in a tent which has 1 tefach (handsbreath) of space above the corpse 3b) if the corpse is immediately above the kohen 3c) if the corpse is immediately below the kohen. The latter two subcategories can best be understood like this, which Jewish cemetaries work to avoid: imagine a corpse buried underground. Because there is no space between the corpse and the earth above him, a kohen who stood above him would be making a "tent" over him (even though there is no space between the kohen and the ground) and become tamei. Similarly, this ray of tumah extend upward an infinite distance until stopped by a tent of some sort (trees work for this). Little discussion that I have seen is devoted to a person under a corpse, but he the corpse were hanging in the air, anyone or anything (this doesn't just apply to kohanim, but is only practiced today by kohanim) passing underneath the corpse would become tamei.
Floors of apartment buildings would work, but the fact that the two apartments are connected by a continuous indoor path complicates things. A building which requires a person to walk outdoors to go from floor to floor (such as some motels & condo buildings) would be less problematic, I think.
Re: Tumah 101
The tumah of a Jewish corpse is transmitted in 3 ways:
Is there a difference between Jewish and non-Jewish corpses?
Little discussion that I have seen is devoted to a person under a corpse, but he the corpse were hanging in the air, anyone or anything (this doesn't just apply to kohanim, but is only practiced today by kohanim) passing underneath the corpse would become tamei.
So a corpse being flown somewhere for burial is potentially a problem. A non-trivial number of corpses are sent to Israel for burial; I assume that dilligent kohanim stay away from the airspace used by planes arriving at Ben Gurion?
Re: Tumah 101
Corpses to Israel for burial:
Additionally, being on the plane taking the corpse would be similarly problematic, depending on how connected the passenger cabin is to the luggage compartment.