I am just curious if there is a country or any state entity in the world where government ministers (equivalent of US department secretaries) are actually elected vs being appointed by the president or legislatures.
- Federal ministers are elected, but by the parliament. The government is not approved as a whole (as it is in many parliamentary democracies) and ministers are not appointed by anyone (and certainly not by the president, who will be chosen between them afterwards and changes every year) but really stand for election, one by one. However, they are not elected directly by the people (if that's what you meant) and in most cases the election is not competitive as the parties negotiate beforehand (it's not always the case however, for example in the last election, one party was promised a seat but submitted three candidates – obviously there is a long backstory but that's not really important here, the point is that there was an election which was not merely a confirmation hearing for an appointment).
- In Glarus (a Kanton or federated state), state ministers were directly elected by the people, gathered in a yearly assembly (Landsgemeinde) until 1971. They still elect the “prime minister” (Landammann) and judges that way. They also vote directly on laws, taxes and a host of other things.