Which properties of planar graphs generalize to higher dimension / hypergraphs?
Oh oh. You want to be very very careful. Contact graphs of convex polytopes in 3d can realize any graph. Surprisingly, the clique can be realized by n polytopes that are n rotated and translated copies of the same polytope (the mind boggles). See this paper:

http://www.cs.uiuc.edu/~jeffe/pubs/crum.html

This already implies that you can encode pretty nasty graphs as intersection graphs of triangles in 3d. See section 4 of this paper:

http://sarielhp.org/p/09/set_cover_hard/

BTW, I am interested in a similar version of your problem by trying to understand how geometric intersection graph behaves...