This game look fascinating, I wish I had the tech to play it. I'm not sure quite what you mean by orbifold, how does this differ from a covering space? I've been playing around with covering spaces for a while and it's possible to come up with a solid implementation in the sense that you can write your engine (or just build on top of an existing one) and then use existing tools to author content. Path lifting does everything for you, though lighting is tricky!
Orbifold is like two euclidean spaces connected with each other. Impossible spaces (or if I understand your term "covering space") are an instance.
There are various things to handle, like AI seeing through portals/doors into other rooms/spaces. I use very simple point lighting - it works good enough but having dynamic shadows could be extremely complicated. Although light maps should be fairly easy.