You are incorrect. I certainly do not assume the government does a great job at running the post office, utility regulation, and so forth. Such a thing is not stated anywhere. In fact, there are numerous problems inherent in these publicly run entities. This still does not detract from the fact that the government becoming involved in such institutions is better than the alternative. However, there are numerous things the government could do to make these programs and organizations much more economically effective (though I'm not really going to go into detail on these).
If the government did not regulate utilities, numerous people would not receive water, electricity, etc, as it would not be cost effective for the companies to provide the pipes or electrical lines to the houses of those who live far from urban areas. Also, assuming such utilities were available, companies would charge these people a lot of money. Also, if utility companies were not regulated, they would be able to use their monopolistic positions to jack up prices to very high levels (the demand for water, electricity, etc is fairly price inelastic, similar to medicine). These are problems the free market has great difficulty in solving, because the extraordinary startup costs associated with going into the utilities business prevents much competition from emerging, and existing companies are able to utilize their position as monopolies to cut prices for a small amount of time, forcing the startup companies out of the market.
In regards to postal service, at one point in our country, this was the only true means of contact between people living in distant areas. If the government did not run the postal service, numerous people at the time would have either not received service (wouldn't have been cost effective) or the costs for the service would have been very high. Therefore, the government felt it necessary to run and regulate the industry. It is more difficult to defend the existence of a government-run postal service now, yet it does still exist. It continues to provide people with a fairly inexpensive form of communication, much cheaper than companies such as FedEx and UPS, especially in regards to smaller articles such as letters, bills, etc, the mailing of which would be prohibitively expensive in an entirely free market system.
You claim that various methods to fund the education of the poorer classes would emerge in the free market to provide them with the means to allow them to attend school. You give the example of citizens directly providing money to those wishing to attend school, apparantly as a form of charity. While it is true that charitable donations have decreased somewhat since the instituion of government welfare programs, enough charitable donations would probably not emerge to replace the aid provided by the government (as judged by the amount and proportion of private charitable donations before high levels of government involvement). Therefore, it seems unlikely that enough money would emerge in this manner to fund the educations of many children. Also, this has to do with the fact that one of the major assumptions of economics is that people act in their own self-interests. Perhaps it is in everyone's best interests that people receive an education. However, such interests are generally not directly seen or acted upon by citizens. There are numerous economic studies examining things such as this, where something is in a person's general interest, yet they do not act because it isn't a direct interest (example: it is in everyone's interest to prevent poverty, as poverty rates are highly corollated with crime rates), yet you do not see high levels of charitable giving or many people voting to increase their taxes for governmental welfare programs). This is because most people think in the economic short run (what benefits am i going to directly see from helping this person?), not the long run, and many feel such actions have a high opportunity cost (they rather act in their own self-interests and get that new car or boat). Certainly there are some people out there who receive some kind of intrinsic benefit from donating money, and that is why they do so. However, this would not be enough to fund education. Also, you say children could get a job working at the school to provide for their education. Such an idea may have merit for those in middle and high school, however it would be economically inefficient for younger students (K-3rd or 4th grade) to do so. What skills would they have, what jobs could they perform that would make such an idea worthwhile for the school? It is doubtful they could even do a good job at janitorial work. It may be feasible to backload their work (free education for the first few years, and make them work hard towards the end of their education to repay it. However, you'd probably see a lot of problems emerge in such a situation (children simply no longer attending the school, which would leave the schools with debts incurred by the students. It'd probably be tough for the schools to recoup their losses from the students in courts too.).
A complete reliance on the free market is generally foolish. It is something hardcore political libertarians believe in, but then can provide scant true economic evidence or arguments to justify. It is true that the "invisible hand" of Smith takes care of so many things, yet government involvement is still sometimes justified.
I imagine my refuting your ideas and arguments could go on for a while, so I'll just stop now. In the future, try not to get into an economics argument with a student earning his PhD in econ at one of the best schools in the country. Hell, most of my professors graduated from the University of Chicago, and even they believe that the government is sometimes economically justified in becoming involved in the economy. Problems emerge because the government often does not do so in the most efficient manner. It is important to find a balance between an economy completely free of governmental controls, and one with extreme governmental control (such as Mexico's ISI programs which existed until the late 1980s/early 1990s, and the USSR).