On the contrary, all of them receive state or federal funding in some way or another.
Actually, the "federal funding" that these receive are all based on local collections of various taxes. So, the money is collected locally, sent to the state or feds, who then juggle it around a bit and hand it back (+ or - a bit).
But "Big Government" is not required for any of these - they would probably be better if "Big Government" didn't insist on handling the money...
No they aren't.
For roads it depends on what Federal or State roads go through a given local area (that comes under a local government). For example, a local government will be paid so much money to maintain and clear (if necessary) a U.S. Highway or a State Highway.
For public schools it varies as to how states fund them. Generally in this state the cities have superior schools to the counties, except in counties with a unified school system (where the cities and county schools have combined). Yet, the state does not handle the property taxes collected by local governments. But we do allocate funding of some of the taxes we do collect. With schools there have been court decisions that have changed funding principles.
Hospitals vary greatly in the type of hospital and the patients.
Electricity varies greatly as well. TVA provides a great deal of electricity here, but is self-sufficient as well. Still, there are many subsidies and tax breaks in the energy sector that make it difficult to determine just how much the Government is helping.