Translation:How many ministers does this government have?
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.
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".