Could you give an example of the other case of de? The only time I can think of it being used as off is with bain...de, which is just how it's said. You really shouldn't expect one-to-one translations between languages, because they rarely happen, especially with prepositions.
Someone made a very helpful comment earlier - prepositions are not always identical between languages. De just does mean both, and you understand from context which is which. English has prepositions which are similarly confusing to speakers of other languages. Hope this helps. (The explanation helped me.)
This is just a guess, but it's probably because it connotes "going off, away from" the thing. So I guess of, off, and from would be denoted by "den", because you are describing something that goes away from the thing. For example, the meat of the beef. when you are describing the meat, you are figuratively "taking it away" from the beef, and discussing it. there's a similar thing in spanish and french.
As far as I know, it is just a case of accent. My grandfather in Northern Ireland would say something like an aspirated 'w', and my grandfather in the south west would say somethig between an aspirated v and hard w.
Don't worry about accent and dialects yet though - its confusing at this stage! Hope that answers the question.
If there's one thing that will never, ever map cleanly from one language to another, it's prepositions. Even when they happen to map semantically (a rarity), they won't consistently map idiomatically.
The Irish "de" can be rendered in English as either "of" or "off" or "from". Context is necessary to decide which it should be.
Part of it is likely dialectal pronunciation bleeding into her recordings. In Connemara, and most of Connacht IIRC, de and do are pronounced the same (as go), as are ag and chuig (as ag), and, when used with an, both sound like gon/don. And, yes, don mhairteoil would be as acceptable without context as this one is (read: barely at all; would need context). As to how you tell which is said, well, it also relies on context. Irish requires a decent amount of context, yet, sadly, DL doesn't give it.