Learn English – Ordering prepositional phrases

One difference between your sentences is that one has an introductory modifier (where the prepositional phrase is in the beginning), and the other does not. This site writes:

A prepositional phrase at the beginning of a sentence constitutes an introductory modifier, which is usually a signal for a comma. However, unless an introductory prepositional phrase is unusually long, we seldom need to follow it with a comma.

Over the edge of the sofa on his fat wrist is not two but three preposition phrases, to wit

1. over the edge
2. of the sofa
3. on his fat wrist

3 (boldfaced) prepositions, each with its own object NP, equals 3 preposition phrases.
It is certainly the case that there are frequent compound preposition constructions with words like edge; but they're no different in principle from compound noun constructions like horse doctor, pony ride, and snake bite.

• [[on/under/over/from/at] [the edge/side/back/arm]] [of the chair/sofa/table/desk]

It is also true that, while the first two PPs do go together as a single constituent in this way, the third one is independently attached to dangling, as a further locative, pinning down Dudley's watch, the antecedent of the relative clause that contains all these PPs.