Are these dangling participial phrases?


I've been spotting what seems to me misplaced modifiers in professional writing, so I would love to get some tips on the following examples:




  1. He'd sit until the last moment, watching them draw near.



Isn't "watching them draw near" modifying he? Then shouldn't it come at the very beginning of the sentence? It looks misplaced, but I'm doubting myself since the sentence was in a story written by a professional writer.




  1. There is the sound of rain on the roof, intensifying, hammering at the gutters.



Is it okay if the participle and the noun are separated by a prepositional phrase?




  1. my spirits are dreadful, owing entirely to the Horrors of every night



It seems to me that in this case, the participial clause is describing the state of my spirits being dreadful. Is it possible for a participial phrase to describe not just one word, but the entire clause?




  1. The number of new UC students from other states and nations will continue to increase this fall, extending a trend that university officials say is financially necessary...



This seems to have a similar usage as above.


Would love any kind of input!




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