Those are my thoughts on it anyway.
Thank you. Your answer has probably been most satisfactory of all answers I've ever had on this subject. It says - as I understand it - that "do you have" is neutral, whereas "have you got" may be emotionally-stricken.
This was in fact a real-life situation which happened to me many years ago in the north of England. My friend, an Englishman by birth, education, residence or whatever else, who was the one searching the key to the entrance door of his friend's house, told me that both versions were correct, but hearing one of them (which one it was, I have forgotten) most Englishmen would get the sense of it, but then would think to themselves: "ha, this chap is a foreigner".
Later on, when I tried to find an answer to this in English language monolingual dictionaries like, for example, the Longman Dictionary of Contemporary English, all I could get was an academic discussion of the problem which was bringing nothing to my understanding of it.
It's an. Because it's any door, you are just giving an example of one. The would be a specific one, either already referred to, or the only one.
That's what I've been thinking, too. But from a certain point of view, this "entrance door" is quite a specific one - "the only one", if I may quote you - it it is the
entrance door of the house to which my friend was searching the key for.