Please do not confuse libertarianism (limited government)
with anarchy (no government.).
Obviously public roads, police and fire dept., drainage and sewer systems, and public building inspectors are all a matter of public safety and use, and as a libertarian I have no problem with those things being managed and regulated at a local and state government level by elected officials, i.e. city councils, city managers, and of course, mayors and governors.
I use the phrase "obviously public roads" because I do believe that limited-use roadways should at some point be privatized. A subdivision whose roads only access other roads in the subdivision and offer no outlet to other public areas should expect no funding from the general public for their roads. In such cases, the land developer and/or homeowners association should be responsible for the maintenance of roadways, and the method of funding, be it an annual fee to homeowners or a toll, should be left up to those associations.
As for public schools - get the government out of the business. End of story. No public funding - give me the money you take from me every year for public schooling, and if I have a child, I'll either send that child to the private school of my choice or home-school. If I don't have a child, there is absolutely no reason for any of my money to go towards the schooling of someone else's. If all schooling became privatized, general scholarships from private organizations would begin to be established immediately, much as those awarded to college students, so I don't believe that this method would deny poor families from getting their children necessary education.
As for the funding of all city, county, and state public-use programs, as long as the money doesn't come from an income tax at any level, I'm game. I fully support the use of state lotteries to generate revenue. I also have no problem with the concept of sales or property taxes at any level. I can choose to buy a new television or DVD, knowing that I will be charged 10-15 cents on the dollar in city and state sales taxes, or even a federal sales tax, or I can choose not to buy said item. If the government weren't automatically taking money from my income whether I spend it or not, I would have no problem paying higher taxes when I do choose to spend it.
Some of your questions may be answered more easily in fictional form. The Probability Broach and The American Zone, by L. Niel Smith, are two books set in a parallel universe with an "America" based on libertarian ideals. The characters explain their philosophies very clearly and make some good points.