Related wikipage: http://en.wikipedia.org/wiki/Gr%C3%B6tzsch_graph
Is the crossing number of the Grötzsch graph known? I have heard it conjectured to be 5 (certainly it is no greater), but came up empty-handed in my search of the literature.
The crossing number of the Grötzsch graph is 5.
Crossing numbers are believed to be difficult to compute in general. (The corresponding decision problem is NP-hard.) However, for small graphs and small crossing numbers, it is possible to find an optimal planar drawing. For example, see Markus Chimani's thesis "Computing Crossing Numbers" for more information.
The Open Graph Drawing Framework (OGDF) can compute crossing numbers. After compiling the program on the linked page and entering the Grötzsch graph, my computer computed that the optimal planar drawing has 5 crossings. Let me emphasize that this technique is exact, not heuristic.