In the first, "Professor Miller" is modifying "Charles" and is not being acted upon - it's an adjective, and a complement to the direct object, Charles, who is being called Professor Miller.
In the second, "the rabbit" is the direct object - it's being acted upon, that is to say, it's being given. Given to whom? Why to Charles, the indirect object of the sentence.
Promise is a very troublesome verb, syntactically speaking.
Most bitransitive communicational verbs, like tell, order, or ask, take infinitive complements with B-Equi from the Indirect Object, thus:
Su told/ordered/asked IO [(for IO) to VP], as in
They told him to prune the peaches.
She ordered him to leave immediately.
I asked her to stop by after church.
These cases are prototype examples of B-Equi, which means that the Indirect Object of the main clause is also understood to be the Subject of the infinitive clause.
They did the telling to him, while he did the pruning. Etc.
With promise, however, the pattern breaks:
Su promised IO [(for Su) to VP], as in
She promised me to leave right after she talked to Bill.
I promised her to take out the garbage.
Here, She did the promising, and she was to do the leaving. In other words, it's the Subject of promise that is understood to be the subject of its infinitive, and not its Indirect Object, as in the prototype cases. This is the pattern for A-Equi, where there is no Indirect Object, as in
- She promised to be home before 11.
- She wanted to be home before 11.
- She tried to be home before 11.
- She managed to be home before 11.
In these A-Equi cases, She is the subject of both the main clause and the infinitive.
It feels like promise is less bitransitive than performative. Performative verbs require an audience, and promise, in particular, requires someone (even if only oneself) to attend to the promise. But that's taken for granted, just like the audiences for say, or swear, or claim, or any other performative verb. The identity of the audience of performative verbs normally need not be expressed in the sentence; this allows promise to be more comfortable with A-Equi and without an indirect object.
Promise also feels more comfortable with a that-complement, when an indirect object is present:
- I promised her that I would trim the magnolias after school tomorrow.
- I promised her to trim the magnolias after school tomorrow.
In a tensed clause, of course, subjects are required and aren't deleted, so no reference problem arises.
Best Answer
Us is here the object of the preposition at, not a direct or indirect object.
The way to tell apart direct and indirect objects is quite simple: a direct object has the verb performed directly upon it, and can be the subject of a subject-verb passive sentence (I saw the man becomes The man was seen), while an indirect object is generally something that receives the DO.
Example: I gave the man a cake. In this sentence, it would make sense to say "A cake was given," but not "The man was given." Therefore, the DO is the man and the IO is a cake.
In your sentence, it would not make sense to passivize it as "We are looked", so the word in question is no DO. Without a DO, an active clause cannot have an IO; therefore, us is neither, instead the object of a preposition.