If you look up なんだか in a dictionary it's translated as "for some reason" or "somehow" but sometimes it seems it's being used more in the sense of "kinda" or "somewhat". How exactly should I interpret it? Is it distinct from なぜか in meaning? For instance, if a person says "なんだか嬉しい", are they saying "I'm kinda happy" or "I feel happy, for some reason"? What about when a person says "なんだかとても嬉しい"? Does なんだか indicate degree of intensity at all?
You shouldn't interpret it "exactly" at all, because it conveys a sense of vagueness, a sense of not knowing quite why or how the current situation came about. It is indeed a conversational filler, and it looks as though you have understood it well. (If by "interpret" you mean "translate into English", then that is a whole different (big!) question -- translating the atmosphere of dialogue is very difficult.)