This answers the first version of the question.This answers the first version of the question. The question has since significantly changed, and continues to evolve.
If, as your title states ("In Inception, How Does Saito Fix Cobb’s Murder Charge?"), you're asking how Saito, in universe, actually solved Cobb's legal problems, that is better asked at Movies & TV Stack Exchange, but as far as I can tell, it is not explained.
If you're instead just asking (as your second-to-last paragraph states) the general question of how one could influence, via "[l]egal plus maybe illegal" means, someone's criminal jeopardy, there are various ways, some of which are more permanently effective than others. For example:
- influence over prosecutorial decision-making, including through bribes or political pressure
- influence over a pardon power (e.g.)