Hi, Anita.
Let's first notice that your examples contain four different kinds of non-finite verbs. The first three belong to a class because their main verbs are all of perception (knew, saw, heard), they all involve a direct object (him, her, someone), and they all involve a non-finite verbal extension:
him to be a spy: infinitive phrase with a copulative verb and predicate noun
her leave the room: a hybrid infinitive/participial construction (neither "to leave" nor "leaving")
someone shouting: present active participle
In those examples, the subject (we, I) simply perceive someone else being or doing something.
The fourth example is different from those. It is "factitive," with a predicated perfect passive participle, "repaired." The subject has caused an action (repairing) to be done (by someone else) to the watch, such that the action resulted in the watch's repaired condition.
Best wishes,
Peter C.
Milton MA