This answer is so pedantic. Why isn't
How many ministers are there in this government correct ?
please tell me what the rules are for when i need to put 'tane'
it is never obligatory but we often use it after numbers, and "kaç" as it is asking for a number.
Actually it is better not to use "tane" when talking about people, unlike this example.
Why does hükümet not have a genitive suffix? I was translating it as "How many minister are there...?" because of the missing genitive suffix.
Does this sense of "government" refer to the entire government (in the American sense) or just to the ruling coalition (European/parliamentary sense), or both?
It's the parliament like your European example. In your words "in the American sense" it is called "devlet".