I think your question is phrased wrong. The free market simply means "voluntary trade" and an "unfree market" means forbidden trade or forced trade. A criminal impedes the free market just as much as government does. **In fact government is nothing more than organized crime that the free market has not yet found a way to stop.**

The question you are looking for is whether or not some people must take it upon themselves to harm innocent people for the benefit of others. Does the end justify the means? Is it ok to kill one person so another might live?  Is it ok to kill one person so that 20 might live?  Does anyone have the right to choose who lives and who dies? Does a group have the right to act in ways a member of the group is not?  

The Free market doesn't solve all of our problems, it just the logical result of adopting a consistent moral framework.

There are so many assumptions in your post that must first be challenged:
1. does the nuclear power plant operate carry insurance, participate in a mutual aid society, or do business with others?  Are others able and willing to shun the operator and those who do business with him?
2. the free market is nothing more than establishing property rights and trading it. This example assumes water can be collectively owned and that the*government* sold the rights.  The free market wouldn't ever allow the ambiguity to exist.
3. people obviously care about their own well being, buying the cheap stuff conserves resources to be put to use in a more productive manner. This argument is a case for "charity" over "growth", ultimately more people benefit from growing economy made possible by high productivity. Over paying because of "conscience" is just malinvestment and ultimately shrinks the pie and makes everyone poorer.
4. people are only entitled to the value they can produce, no one is entitled to enslave a teacher. Someone born to a poor family is able to learn everything they need to succeed for free.  Learning is done by the student, not pushed from the teacher.  All you need is time and motivation.
5. companies compete against each other for labor, anyone who actually runs a company knows that employees are subject to supply and demand just like jobs are.     This is just another way of stating #3.
6. this is just another way of stating 5 and 3.  In this case what is being sold is surplus meet rather than surplus labor. 

In effect most of your arguments boil down to a personal disagreement with the free market established prices. In some cases the distortion in prices is being caused by the governments and going to a free market would help, in other cases adding government to 'fix' the problem would create many other problems which you have failed to account for.  It all amounts to price fixing which causes shortages and gluts in the market. This in turn results in economic inefficiency and makes everyone poorer.

Some of these are easier than other:

1) I have no idea how a free market could control nuclear materials to the same extent the law has
2) Why is this bad? If someone chooses to pay for the cleaned water instead of cleaning it themselves... it's probably more expensive for them to clean it themselves!
3) Animal welfare and ecological welfare is tough. Working conditions are a simple matter of improving coordination mechanisms for collective bargaining.
4) Does being a human entitle you to enslave the brightest minds in the world and force them to be educators instead of researchers? It's a spectrum. I bet in a different thread you'd argue expensive educations are not necessary because the internet has infinite learning resources.
5) The basic argument against minimum wage is that it discriminates against unskilled labor. If you're going to introduce deadweight loss to be able to support the least productive, use universal basic income instead - more straightforward and we don't force people to work just so they "earn" money they didn't actually earn.
6) Just to be clear, you're suggesting that either a) the food should be thrown away, or b) that it should be sold at an artificially high price? One interest group sees their income source destroyed, another sees cheaper fish to feed their family.

I've sometimes jokingly said that I'm anarchohierarchist – meaning that I believe we need hierarchies, but every individual should have freedom to choose which one is most suitable for them, or to set up their own.

But usually I just call myself a utilitarian voluntarist with a basis on talebian antifragilism and public choice theory.

My answers:

1. Depends on what kind of agreement there is and what the overall situation is. Because nuclear waste is really dangerous, company should have done some plans beforehand and also make sure that they have funding. If they didn't do that, police could stop operation and maybe sell it to other company.

	With freedom comes responsibility. With nuclear waste the responsibilities are very big. If it seems that those responsibilities aren't taken care of, state (or other hierarchy) has a right to step in and make sure that situation won't develop in a way that will be harmful for other people.

2. Water is a scarce resource, so making people to pay for it is not usually a bad thing. It will improve both availability and quality.

	To think what has happened in Africa isn't very good example in this case because there are not much free markets in Africa.

	I'm not against locally governed water supplies, but those are not very practical solution nowadays. It usually takes several generations to find out how water supply can be used in a way that everybody considers it to be fair. Societies are changing so rapidly now that this is becoming more and more difficult.

3. Environmental problems are tough problems in every system. If people don't care about environment, they will vote politicians that don't care either. AFAIK the best solution is to have rich citizens who can afford to donate for environmental charities and who can buy sustainably produced food and other products.

	If people are poor, they don't usually think how the food they are eating is produced. They will eat anything to survive. This is why socialism is usually really bad for environment.

	Now that we have internet, spreading important information about working conditions isn't really that hard. It's easy to generate a shitstorm against employers who treat their workers like carbage.

	"Low regulations for working conditions" is usually problem only in systems where the degree of free market is also low. If people are poor and they don't have any other possibilities, of course it sucks to be a slave of evil employer. But in a free market system there is usually plenty of other choices where to work, or to set up own business.

4. Good education is really, really hard to mass produce. Usually politicians and bureaucrats don't have any idea how to do that. Of course they will try, but the result is bad quality for students and high price for tax payers.

	I live in Finland where the education system is supposed to be best in the world. But in reality kids don't learn much in there, mostly just how to pass a test. Problem is that in real life adults don't have to pass tests.

	Skills and knowledge how to live a good and productive life are not taught. Youngsters and young adults have no clue how to operate in real life when they get out of the education system. They have pretty much wasted several years of their childhood.

	Nowadays information is so cheap that poor families don't have to worry. Price of real education isn't a problem anymore.

5. If employer pays too little, employee can always get a new job or set up own firm. In a free market job creation isn't prevented so most of the time there will be lots of available jobs. This means that payments will be based on worker's productivity, not on an absolute minimum that worker can survive.

	There is always more supply for jobs than demand. People will always want more stuff and services. A large number of unemployed people usually means that this natural state has been disturbed by really bad regulations.

6. Food dumping is usually a consequence of food subsidies. Governments pay for overproduction and then they have to find a way to get rid of excessive food. In a free market food production is much more aligned with the demand for food so massive surpluses aren't created.

	Overfishing is the result of difficulty to own water systems. If there were a way to own water as we own land, the problem would be way smaller (same applies to air). Best way to solve it is usually that the state (or other hierarchy) auctions the rights to fish. It can regulate the quantities so that nothing gets near extinction.

Free markets would be much better for society and is an ideal to strive for, but it won't solve all our problems.  A freer market would just minimize problems, but bad actors and violence would still exist and conflict would still need to be resolved.  In the end it's just a matter of what people think is the best way to resolve them.  Humans are fallible and there will always be problems.  This may also be another reason why truly free markets may be hard to achieve.   

It's difficult to assess what an individual's beliefs are either today or in the past with just labels.   It could be argued that the younger generation's current beliefs about anarchism are different from  what Kropotkin, Proudhoun, Nestor etc actually believed.  It could also be argued that Proudhon/ Kroptokin/Nestor could be considered 'libertarian' and believed in voluntary free markets at the local & micro levels.   Whatever the case may be, it would be good to read their writings directly.   It would also be good to compare each anarchists stated beliefs about Marxism and compare them to what a modern day 'left' anarchist would say.  What did Kropotkin/Proudhoun/Nestor believe about Marxism?  How does that compare with a modern day 'left' anarchist?  

In any case, labels and definitions make it difficult to compare.  I would consider Lao Tzu/Lysander Spooner/Benjamin Tucker/Henry David Thoreau anarchists.. and would even consider Thomas Paine & Patrick Henry and the anti-Federalists more anarchist...  they can also all be considered libertarian.   If you look at Wiki Kropotkin was influenced by Proudhon, economist John Stuart Mill and Godwin who was influenced by the early Edmund Burke (the father of modern day conservatism/limited government/classical liberalism/libertarianism and just a few degrees away from ..yes anarchism.... so much for labels!).   Anarchism is just the logical extension of the ideas of limited government, that is the absence of government.  

So where does regulation and the solutions to the problems you listed come from?  If you say government, what do you think all those 'anarchists' mentioned above believe about regulation and the problems you listed?  

If you say 'regulation' or a solution can come from outside of government then I think it would be closer to what 'libertarians' and the 'anarchists' you mentioned think.   Anyways good questions.. good luck on your search for your answers.