reference request - Computing the index of a Lie algebra: what is known beyond the reductive case?

There is quite a bit of literature by now, in the classical characteristic 0 setting of finite dimensional Lie algebras. Looking up some of the papers listed below on arXiv (usually under math.RT) and others they refer to would be a good way to get into the recent work, including some on nilpotent Lie algebras. Beyond this, I can't answer your specific questions in detail. But as Victor points out, study of the index is only one step. Even in the reductive case, the rank is just one piece of information.

Dmitri I. Panyushev http://front.math.ucdavis.edu/0107.5031

A.N. Panov http://front.math.ucdavis.edu/0801.3025

Jean-Yves Charbonnel and Anne Moreau http://front.math.ucdavis.edu/1005.0831

Celine Righi and Rupert W. T. Yu http://front.math.ucdavis.edu/0908.4201