We often prefer more specific placements in Norwegian than just being,er, stuff is hanging, lying, standing, sitting...
If you asked someone for directions "where is the hotel", Hvor er hotellet? there'd be a fifty-fifty shot of the answer being either ligger or er. But for a hotel, it is generally much more common to lie somewhere. Hvor ligger hotellet? is more about a description of the surroundings of the hotel, than directions on how to get there.
I think that for a really tall structure like the Eiffel Tower you could choose, but my guess is most people say "ligger", that's what feels most natural too me. "Står" adds more meaning than merely the location, even for something tall.
(On a side note, raising a structure, reise et bygg. The Eiffel Tower was raised in 1889. Eiffeltårnet ble reist i 1889.)
It refers to the location of the hotel, not a property of the hotel/building.
The only time I see "standing" as a property of a building, is in connection with building or demolishing/destruction. If a bomb goes off near a hotel, you can say "hotellet står ennå" to describe the fact that is wasn't damaged enough to collapse (or not damaged at all), the hotel is still standing/the hotel still stands.
'by the airport' sounds very natural to me. 'next to the airport' has the same meaning, but 'near the airport' sounds like it could be farther away than the other two.
Maybe 'next to' is not accepted because duolingo wants you to learn the literal translation of 'ved', since I'm pretty sure you'd say 'next to' in a different way.