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, The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser, https://arxiv.org/abs/math/0107031, https://doi.org/10.1017/S0305004102006230
A.N. Panov, On index of certain nilpotent Lie algebras, https://arxiv.org/abs/0801.3025
Jean-Yves Charbonnel and Anne Moreau, The index of centralizers of elements of reductive Lie algebras, https://arxiv.org/abs/0904.1778, https://www.math.uni-bielefeld.de/documenta/vol-15/11.html
Celine Righi and Rupert W. T. Yu, On the index of the quotient of a Borel subalgebra by an ad-nilpotent ideal, https://arxiv.org/abs/0908.4201, https://www.heldermann.de/JLT/JLT20/JLT201/jlt20005.htm
