Translation:Noord-Brabant does not have its own government.
it's not a /regering/, but yes, it does have a local government. that'd be called Provenciale Staten
It depends. In a unitary state it does indeed only refers to the national government, but in a federal state like Belgium it also refers to the governments of the regions and communities. So in Belgium there are, among others, the 'federale regering' (federal government) and the 'Vlaamse regering' (Flemish government). I think a general rule of thumb is that a 'regering' has ministers.
"his/her/its own" is usually the best translation of the Dutch word "eigen".