If an accident were to occur at a cross-border airport, who would investigate?
By ICAO conventions, the lead investigator is the country in which the accident/incident occurs, unless otherwise delegated. This is fairly common: for example an engine fails in Canadian airspace but they divert to Iceland, it's initially Canada's responsibility.

So you have an argument for whatever country the airplane hit the ground first in your example of a hard landing.

Additionally, ICAO Annex 13 recommends that if the country can't be decided, the state of registry should conduct the investigation:


  5.3 When the location of the accident or the serious incident cannot definitely be established as being in the territory of any State, the State of Registry shall institute and conduct any necessary investigation of the accident or serious incident. However, it may delegate the whole or any part of the investigation to another State by mutual arrangement and consent.


Practically, I would imagine they would delegate the investigation to the country of registry of the aircraft, since that's where the tax dollars were paid.