The premise of the question, namely that "we need a direct object to form a passive sentence" is not correct. Active sentences with prepositional phrases can indeed be converted into passives, such as in the first example:
I am being played with (by him).
Google shows plenty of hits with the similar phrase "You are being toyed with".
In fact, all of the sentences listed can, in an exercise in syntax, be converted to the passive as follows:
I am being sat by by him
The bed was slept in by me
School was arrived at by Mary
The ground was fallen on by him
Whether such constructions are considered acceptable has a lot to do with why we use the passive in the first place. Clearly, the passive allows the speaker or writer to make a certain person or thing the subject of the discourse.
So, the active sentence:
The decorators arrived at the school shortly before dawn
could in theory be converted to:
The school was arrived at by the decorators shortly before dawn, and by late evening had been completely repainted
if we wish to make the school our focus, not the decorators.
No doubt the passive here would still be found questionable by some. And this may also have something to do with the greater acceptability of idiomatic verb + prepositional phrases in the passive. Compare the following two sentences:
The room has been gone into many times today.
This problem has been gone into many times.
The second sentence with its idiomatic use seems much more acceptable.
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.
Best Answer
Our esteemed professor has in a potentially ephemeral comment above written this fine answer, which I herebelow consign to posterity and the general community pro bono publico: