That is, by adding a handle one can save on the number of crossings. To wit, consider a large plane triangulation T with two vertices u and v far apart. Adding in the edge uv makes a toroidal graph with large planar crossing number. One might expect that that is the desired characterization. Adding a handle can save more crossings than the previous handle, but not by an arbitrarily large ratio.
Every sequence is a crossing sequence.
There is an obvious analogue about the crossing sequence for graphs embedded on non-orientable surfaces. This supports my bold conjecture. Jozef Siran introduced me to the problem. To my knowledge we are the only people who have worked on it.