There are lots of results provable in this context. In the book I wrote with Heiner Kamps (which is easily found via Google so I won't advertise here!) we looked at the problem of what results in homotopy theory could be proved with a restricted set of fillers for boxes in a cubical enrichment of a category.  This applies to your question since groupoid enriched categories give rise to such cubical homotopy theories very easily.  

There is an old paper: P. H. H. Fantham and E. J. Moore, [Groupoid enriched categories and homotopy theory](http://dx.doi.org/10.4153/CJM-1983-022-8), Canad. J. Math., 35, (1983), 385 – 416, which also examines this question and of course, some of the classical book by Gabriel and Zisman is devoted to developing GE-categories in your sense.

As Noah points out, these 2-categories are nowadays more often called (strict) (2,1)-categories although that term (without the `strict') also is used for bicategories in which the homs are groupoids. Try looking up locally groupoidal 2-category in the nLab for more on that side of things.

(Edited (08-01-2018): I should have mentioned the extensive work by Hans Baues and his coworkers on what he calls 'track categories'.  These are the 'ge-categories' of the question. There are many problems solved within the more calculative part of homotopy theory that are stated in terms of these track categories but which have direct interpretation in more classical approaches homotopy theory.)