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It would be better if someone familiar with the Longman book responded
to your question. I am not familiar with that book.
Nevertheless, I will try to answer your questions.
1. An object of a preposition always follows a prepositional
verb, yes. A prepositional verb is a combination of verb and
preposition. The preposition is followed by its object.
However, an object of a preposition does not
only follow a
prepositional verb. Prepositional phrases, that is, preposition
and noun, can be found in other locations within a sentence. If
the phrasal verb is intransitive, nothing need follow it. If it
is transitive its direct object follows.
2. I believe that "Object of a Preposition" and "Complement of a
Preposition" are simply two different terms which mean the same
thing. According to the quote you provided from your book, the
entity referred to by both of those terms would be an oblique object.
3. Neither example given is passive, so I'm uncertain what you are asking regarding the passive.
In my opinion, the standard terminology says that there are direct
objects, indirect objects, and objects of prepositions. The
object of the preposition "to" can be an indirect object and an object
of a preposition at the same time. Otherwise the three categories
are mutually exclusive.
Paul won the prize. ("the prize" is a direct object.)
Paul gave Mary the letter. ("Mary" is an indirect object. "letter" is a direct object.)
Paul gave the letter to Mary. ("Mary" is the object of the preposition "to"
and is an indirect object. "letter" is again a direct object.)
Paul traveled to Italy. ("Italy" is the object of the preposition "to"
but is not an indirect object.)
The man with the black hat drove his car into the garage. ("hat"
is the object of the preposition "with". "garage" is the object
of the preposition "into". "car" is a direct object. Note
that "with the black hat" has an object of a preposition, but it is not
positioned after a verb.)
I believe Longman wants to classify these different kinds of objects
differently. He wants only two categories: direct objects
(with the same definition as seen in the examples above), and oblique
objects (any
other kind of object).
CJ