Well, okay, obviously if you have wood paneling (BE also: wainscoting), there's some structural wood behind or underneath it, so I suppose the general concept includes that, in a way. However, to me that's really not the primary/actual meaning.
My dictionary translates Balkenwerk as timbering, beams, which are not hollow, so they couldn't have rats inside them. I was assuming you were using it more in the sense of structural wood inside the walls, which we might just call two-by-fours or studs nowadays (in AE). But even there, the rats would not be inside the wood supports themselves, but inside the walls, which is slightly different, isn't it?
I still don't think we normally mean hidden structural wood when we speak of woodwork, but rather the wood that's actually visible in the room. In a modern house it's often now only with reference to painting, as in the NOAD definition, which mentions window frames and doors. It could include things like exposed ceiling beams, but that too would be mainly historic.
What I'm saying is that when we think of rats scurrying in the woodwork, or something coming out of the woodwork, we think of noises behind the walls, which used to be paneled. If you had a brick or stone house, with sheetrock or plaster walls, you probably wouldn't speak of woodwork except with reference to window frames and doors.
I think. See what others say.