I agree with BillJ's first two comments: being ill in the first example and her father in the second are integrated well into their clause structures and hence do not qualify as supplements in the intended sense. They are, rather, post-dependents of the preceding noun -- on a dependency grammar analysis. I would analyze the first sentence as follows:
And the second sentence as follows:
In each of these structures, the subject noun takes a post-dependent, the post-dependent clearly modifying the preceding noun. The arrow dependency edge in the tree each time identifies the post-dependent modifying the subject noun as an adjunct; it is not a complement.
The third example is, I think, indeed a challenge to strict head-dependent analyses. Phrase structure has a means of addressing such cases, though. Phrase structures can produce exocentric structures, that is, structures that are headless. Hence I would attempt an analysis of a washer-dryer along the following lines, combining both dependency and phrase structure:
The horizontal connector above washer and dryer links the two words in such a manner that neither is head over the other. In this manner, an exocentric phrase structure is integrated into the greater dependency analysis. The analysis of the fifth example, i.e. paperweight-cum-ashtray could receive a similar analysis.
A related point concerns the analysis of coordinate structures. One can legitimately argue that coordinate structures are headless (or they have multiple heads). This suggests again that exocentric analyses are necessary. I have therefore argued that the following type of analysis is appropriate for coordinate structures:
This analysis links the nouns together without viewing the one as head over the other.
In sum, I agree with the underlying sentiment of the question, namely that strict head-dependent analyses are not always possible. My solution to the problem is to acknowledge exocentric phrase structures at times, and then to integrate these exocentric structures into the greater endocentric analyses of sentence (and word) structure.