Perhaps it’s unfair to ask these sorts of questions of kids’ books, but … :
(1) If the Minister of Magic can be sacked, and a new one appointed, who does the sacking and the appointing? Is there a wizard sovereign and/or a wizard parliament in J.K. Rowlings’ world? Is it a constitutional monarchy, with regular elections, or is it some kind of oligarchy?
(2) How are government services paid for, like salaries of people in the Ministry of Magic and or teachers at Hogwarts? Even if there are school fees, I doubt if they are high enough to pay the costs of Hogwarts, given that a lot of students like Harry Potter get a free education. And wizard money is not in unlimited supply, since if it was, you would not need Gringotts Bank. So is there some sort of tax system in the wizard world? And (given that he is relatively wealthy in that world), does Harry Potter pay taxes?
(3) Who exactly decided that top-level dealings between the wizard world would be between the Muggle Prime Minister and the Minister of Magic? And was it the same 400 years ago, when the Muggle PM was less important, and the Muggle King/Queen of England was far more important? If there has been such a change, how did the change take place? And what happens in other countries? Does the Minister of Magic in the US deal with the President of the US, or with someone else?
I know a lot of these questins aren’t answered by the HP series, but it’s interesting to spoeculate.