Is "door" the direct object of "The cat ran out the door"?

I think you are right.

'out of the door' seems like an adverb.

If it was 'the cat ran a race' then 'a race' would be the object.

I also think you're right about the cat being an object. Often people say in a sentence like that the verb is intransitive - it has no object. But others have long pointed out that the cat ran itself out of the door.

A traditional definition of "direct object" is that it says what receives the action of the verb. The verb gives some action or event, and whatever the direct object refers to is affected by that action or event. By that test, in your example, "the door" is not a direct object, because the door needn't be affected by the cat running out of it.

The existence of a passive is also a pretty good test for direct objects, as you mentioned. If "the door" was a direct object in your example, you'd expect a corresponding passive "The door was run out by the cat". This does not sound very good, but a related form sounds much better: "Which door was run out of by the cat?" However, this can probably not show that "the door" is a direct object, since there is a construction called "prepositional passive" that makes the object of a preposition into a subject, rather than making a direct object into a subject.

There is no object in that sentence. The prepositional phrase "out the door" is an adverb describing ran.

"to run" is a verb of movement, a verb class of its own, even if dictionaries only have meagre two verb classes (construction classes: transitive or intransitive). Verbs of movement normally have no direct object. Mostly they are followed by a where-to indication (destination). Also description of the course can follow such as to march through the woods/ across the fields. "The cat ran out of the door/AmE out the door" indicates the course the cat took. I would not say "out the door" is an adverb. In any case it is no direct object as it is a word group with a preposition.

It is a specialty of English that verbs of movement (vmov) can be constructed transitive (with a direct object). Then the vmov is followed by a noun/noun substitute as in "Hannibal marched his troops/them over the Alps into Italy". In other languages I know transitive use of vmov is not possible.

As rogermue indicates in his answer, verbs of movement like run can actually have an object.

So let's introduce an actual object in that sentence:

The cat ran the mice out the door.

I think that it is very clear now that we have:

• a subject (the cat), which performs the action
• a verb (ran), which describes the action
• an object (the mice), who underwent the action
• an adverbial prepositional clause (out the door), which modifies ran, indicating where (or rather _in which direction) the action took place

If you are in doubt that the mice is an object, you can do your little passive-voice test:

The mice were run out the door by the cat.

Now the question is, when we remove the object, does an adverbial clause that happened to be in the sentence all of a sudden an object?
I see no reason why it would.

I ran a test yesterday.

If I remove the object (clearly, a test) from this sentence, sure, the meaning changes:

I ran yesterday.

But in both cases, yesterday just tells us when the action took place, whether I performed a test or an atletic feat. It does not become an object for any reason; why would an object tell me when something happened, anyway?

There is no direct object in the sentence "The cat ran out the door," "out the door" is a prepositional phrase functioning as an adverb. If you change the sentence to read "The cat ran the cat out the door," you do get a direct object, but you then change the meaning of the sentence. The cat then would be willing itself to chase itself out the room.

The simplest test is syntactic: direct objects are never prepositional objects.

This is not to say that direct objects can't (appear to) follow prepositions, but in all cases it's because the preposition is actually a verbal particle (a la German's separable prefix verbs). To test this, see what happens when you move the preposition to the end of sentence:

The cat ran the door out

is nonsense at best, since it implies the cat caused the door to leap from its hinges and leave the house (or whatever building this is happening in). In any case, it changes the meaning drastically. Now compare

The cat ran out the mice.

This is a valid sentence, and "the mice" is a direct object, since

The cat ran the mice out

is also valid.

All this to say "to run out" is a valid verb with a transitive construction (two, in fact), but it is not the verb in your example.

And if your friend tells you "you can't end a sentence with a preposition", tell her, "you're right I can't, which is why 'the door' isn't a direct object".