The police is called “polismyndigheten” which is a governmental organisation where all police staff (both civilians and no civilians) are employed (with some exceptions like the “security police” which is a fully separate organisation).
And no, a police from the Stockholm police won’t patrol in a city on the other side of the country. There are police districts and such but it’s still the same organisation. You could probably get transferred to the other side of the country, but that would obviously be a bad idea unless the employee is moving anyways. I suspect the union might have something to say if they decide to transfer people wherever for no reason.
Except in informal settings the police is only called the police. For example police cars only say “Police” and never “Stockholm police” or similar.
But there are obviously some specialised divisions/groups within the organisation that are only present in one place (probably Stockholm). A very small police station in a small town could very well be investigating murder but I suspect they will likely want help from some other station or specialist division if the case is complex.
I haven’t seen one in a while either, but looking at images it does indeed appear that there are no region markings on the cars.