It's a State Policeman, but in Italy there are no States, they are called provinces, so they have Provincial Policemen.
The most correct translation is Provincial Policeman, but to make an american, or anyone else from a country that also doesn't have Provinces but States, you can translate as State Policeman because it's basically the same thing.
Yes, but it would sound unusual. In Italian, usually the already known information comes before and the new information afterwards. So "un poliziotto provinciale arriva" implies that "un poliziotto provinciale" is known and that the fact of arriving is the new information. On the other hand, "arriva un poliziotto provinciale" implies that the fact of arriving is already know/less important and who arrives is the newest/most important information.
Reference: https://italian.stackexchange.com/questions/11269/arriva-un-poliziotto-provinciale-vs-un-poliziotto-provinciale-arriva (read the first comment)
Because Country Policeman is wrong. Just because in some other sentence it makes sense to translate as country it does not mean it works in 100% situations.
The only situation that I can think of where Provinciale can mean Country is Via Provinciale because then you can translate as Country Road, because they are very similar things.
As I explained below, Provincial Policeman is basically the same as State Policeman, it's just that Italy has Provinces and not States.