First of all, Nate, the orders of words in English are not fully prescribed by a set of rules. Some orders create confusion as to meaning or fail to convey a clear meaning at all.
Behind the several differences the open door are hidden
Here there is an obvious mistake: the main verb is plural where it should be singular. Even when this 'mistake' is corrected and are is replaced by is, or door is replaced by doors, the 'grammatically correct sentence makes no immediate logical or empirical sense.
But your alternatives all make perfectly good sense. And you are right to think that, if we are looking at it in terms of how sample sentences are usually set out in text books for learners of the English language, your version would be the standard version. Except, of course that this particular sentence is more literary than the type that would appear in English language text book, at least at the earlier stages.
Some variations are absolutely wrong, of course. You cannot reverse subject and predicate.
A car was run over by a stray dog.
This is perfectly grammatical, of course. Simply, we know it cannot be what the news reader meant to say. In the case of your sentences, the order depends on what is being emphasised. The 'open door', is obviously a metaphorical reference to something like a negotiation between two parties. The 'open door' suggests that outward willingness on the part of one or possibly both parties. The point of the sentence is that despite the apparent willingness 'several differences' matters may not be a be as easy as they seem.
The order creates suspense by making us wait for the 'punch line'. "Behind the open door" <"What's going on behind it?", we are thinking>. "... are hidden" <"what?, come on, get on with it!" "...several differences." This, in turn, makes us alert and ready for something very important.
So the rhetorical effect is to build tension towards a climax.