What are some potential approaches to classifying/characterising (finitely presented?) groups that act (without fixed points and potentially more conditions) on CAT(0)-cube complexes? As so many groups act on CAT(0)-cube complexes, is this sort of classification/characterisation even possible?
The question is probably too broad to get an answer, but at least Niblo and Roller's article Groups acting on cubes and Kazhdan's property (T) should be mentioned. Therein, they proved:
Theorem: A group acts fixed-point-freely on a CAT(0) cube complex if and only if it contains a codimension-one subgroup.
(To be precise, they proved this statement with connected cubes instead of general CAT(0) cube complexes, but a CAT(0) cube complex always embeds isometrically and equivariantly into a connected cube, so the two statements turn out to be equivalent.)
The motivation comes from Sageev's thesis, where he characterised codimension-one subgroups as subgroups arising as stabilisers of essential hyperplanes for actions on CAT(0) cube complexes.