"One of the first questions would be- What services and what production should be government run?"

I would say the govt should be in charge of any and all business to do with war making.

That govt should be in the exclusive business of providing justice in the form of courts, the fabrication of laws, and the enforcement of these laws.

The govt should also be charged the the task of managing a national monetary system- the purpose of which is to provide a medium which is universal and uniform and has the confidence of being guaranteed by the government.

The govt should also be in the business of managing borders which includes managing the traffic of people and goods.

The govt should have the task and duty of responding and resolving emergencies, from such thing as house fires, and natural disasters.

The Govt should have a monopoly of the business of using physical violence to resolve matters- as long as that balanced by a govt which for the people, by the people, and of the people- which involves such things which allow this to occur: a free press [free from governmental control], right to assemble, etc.

As far as a governmental health care system. I would suggest one first try have governmental animal care system. And if a government can provide a fair and good service of providing health care to all the pets, cows, or whatever, then perhaps they can try it out on humans.

The govt business in education is to insure it's citizens understand the form of government they have- it's history and how it works.