The first question is always: is this information in OpenStreetMap.
For the name of the island, it probably is.
For the Bayrischer Wald, I doubt it, as this is a region without well defined boundaries.

Then if it is in OSM, you need to know how it is mapped. Rügen is probably not mapped as an administrative region, so it will not show up in the “stack” you get for a given coordinate (afaik).

So I fear that the current implementation of Nominatim cannot return the answers you are looking for.
Perhaps someone else will prove me wrong.