with [noun phrase], S+V+O.
S+V+O, with [noun phrase].
S+V+O with [noun phrase].

Which objects do "with [noun phrase]"s try to modify?
Is it true that "with [noun phrase]" always logically modify the closet noun?
I *think* that I have seen an OG question (with the same structure in my example #2) with an explanation complaining that "with [noun phrase]" could logically modify the subject or the object of the preceding clause.

Thanks in advance.

this is one of those constructions that drives you batty: the 'with' construction can function as either an ADJECTIVE modifier or an ADVERB modifier.

examples for the first construction:
with teary eyes, lindsey told her mother about the broken vase. --> 'with teary eyes' is an ADVERB modifier, describing the action of telling her mother
with 34 million inhabitants, california is by far the most populous state --> 'with 34m inhabitants' is an ADJECTIVE modifier, modifying 'CA'

same things can go for the second construction.

--

the last construction is slightly different rhetorically, but the 'with' construction can still serve as either an adverb modifier or an adjective modifier:
i draw designs with a permanent marker --> 'with a permanent marker' is an ADVERB phrase, modifying the action of drawing the designs
i adore the girl with the big blue eyes --> 'with the big blue eyes' is an ADJECTIVE phrase, modifying 'the girl'