Which properties of planar graphs generalize to higher dimension / hypergraphs? A planar graph is a graph which can be embedded in the plane, without having crossing edges.

Let $G=(X,E)$ be a $k$-uniform-hypergraph, i.e. an hypergraph such that all its hyperedges have size k.

There has been some work done on embedding hypergraphs in the plane (with the context of clustering or some other application), but often, the data just can't be embedded in the plane. The solution could be either to force it, with some loss, or embed it in higher dimension as I suggest here:

A natural extension of planarity (IMO, at least) is a "$k$-simple-embedding" (is there a known different name for it?) of $G$: an embedding $\mathcal{M}:X\to \mathbb{R}^k$, such that there exist surfaces which connect all of the vertices of each hyperedge, and these do not intersect except for the endpoints.

(Think of the analogue in 2D, where each surface is an edge you can draw however you like).

Here's an example of a valid 3-simple-embedding of a 3-uniform-hypergraph. (Each vertex is colored by the hyperedges it is contained in, and each face represents an hyperedge). 



Another example of 3-simple graph is the complete 3-uniform-hypergraph on 5 vertices $G=(V,V\times V\times V)$. To see this simply take 4 points in $\mathbb{R}^3$ which don't lie on a 2D plane, create a triangular pyramid (their convex hull), and place the fifth point in the center of the pyramid, connecting it to the other vertices.

Similarly, it seems that the complete 3-uniform-hypergraph on 6 vertices doesn't have a 3-simple-embedding.

There are some very useful properties of planar graphs which allow improved algorithms for hard problems when the graph is planar. Unfortunately, the data is often not planar, although sometimes it is of low dimensionality. I think that understanding which properties of planar graphs generalize will help us figure out which algorithms can be adapted for higher dimension with the same tool.

An example of a property that could be useful comes from Fáry's Theorem which suggests every planar graph can be embedded in a way that all of its edges are straight line segments. 

Does Fáry's Theorem hold in higher dimension?, i.e. if a graph has a $k$-simple-embedding, does it have an embedding in which all of the hyper edges are hyperplanes?

Are there any other properties that can be generalized? for example, can Euler's Formula for planar graphs be generalized somehow to higher dimension? (although at the moment I'm not sure what would be the meaning of it).
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...