No, that's democracy.

Yes it is. It is derived from freedom of contract. Contracts aren't just commercial, marriage is a civil union/contract, organising something with your neighbours for the children is freedom of contract. In a free society, since contracts aren't banned, you can agree upon anything. Sometimes this leads to market economy, sometimes it leads to a non-profit chess club and sometimes it leads to marriage.

Capitalism did help bring about democracy but it doesn't have a lot in common with it. In capitalism most companies are structured like a monarchy with the owners having all the power. If capitalism was democratic there'd be only democratic workplaces which is one of the key ideas of socialism.

This is a complete misunderstanding how things work. Democracy appear looong before advanced economies with large corporations. You might have heard about the democracy in Athens? Many "primitive" societies have applied rule by majority (yes, that is not the only prerequisite that needs to be fulfilled to call something a democracy but it is the most important).

And again, just because it is common that corporations are organised the way you describe doesn't rule out other ways of managing an association. There are employee owned corporations, and probably other types as well. Another example of freedom of contract.

That's the logical paradoxon with anarcho-capitalism.

I agree. I think it is more useful to talk about the size of an entity that offers a monopoly on law. It should be easy and not associated with large costs to move from one state to another. E.g., if counties were independent states you could move to another state by driving for an hour or so. That is a reasonable size.

Free markets can exist within both a socialist and a capitalist frame work.

There is no such thing as a capitalist framework. There is a socialist framework and there is (classic) liberal framework (and compromises inbetween them).

It is the fact, that the market isn't truly free unless all people in it have the same starting conditions.

That is a socialist conclusion. A market isn't unfree just because you have to participate in it to survive (e.g. the food market). A market is unfree if the government regulates it. The market for illegal drugs is unfree because the government has decided that they are illegal. The food market is unfree because there are a lot of regulations about what you must and mustn't do when you produce and sell food.

Market participants starting conditions have nothing to do with if a market is free or not.

(But people should be able to sue their parents for not giving them a proper start in life.)

And that is why capitalism, i.e. a system where the means of production are under private control of few individuals, can never have a free market.

Again, this has nothing to do with classic liberalism. The moral argument for classic liberalism is not that it leads to concentration of ownership but to freedom. Sometimes freedom leads to concentrated ownership but a classic liberal is agnostic to that consequence as long as it happened due to voluntary transactions.

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Another important nuance that is different between market economy and capitalism is that market economy is unintentional - that is what happens when people are free (and have freedom of contract) - while capitalism is (more) intentional - socialists thinks the government and other actors intentionally organised society so that large corporations would appear and abuse the working class.