(1) each individual should be rewarded by society in proportion to what he does for society, rather than by his possestions, descent, titles, intelligence, shrewdness, etc.;
That's what capitalism does. If you make a profit, it's because you utilized your capital in a way that society a.k.a. the market values. You get market share by giving customers what they want. You make profits by doing so efficiently.
Not at all! Capitalism (as a political term) basically says that whatever you can grab by following the rules of the game, you can keep. In particular, it sees no difference between gains from speculative trading, monopolistic and abusive pricing, deceitful marketing, exploitation of cheap labor, activities that damage the envirnment or public health, etc.; and the state should not try to hamper such activities. In Capitalism, poverty and inequality are non-problems.
(3) the state is supposed to provide public services like health care, education, social security, transportation infrastructure, emergency and security services, etc.;
and the State gets the resources to do this how exactly? By running a bake sale? If a private organization takes things involuntarily, it's robbery. Just because the State calls it "taxation" doesn't mean it's any more moral. ...
By taxes, of course. Socialism generally implies higher taxes, and progressive income taxes -- to counter the "rich get richer" consequence of capitalism.
But hey: I am not trying to convince anyone here that socialsm is good. Just trying to explain what "socialism" means.
Free markets by definition are free from State interference. Monopolies are only possible with state help. Cartels don't work, witness OPEC.
That is a serious distortion of the term "free market" that Libertarians and Anarchists have invented. Sorry,
a free market is totally not a market that is free from regulation and control.
Basically, it is a market where consumers are free to chose among suppliers, suppliers are free to set their prices as they like, there are no artificial production quotas, and -- most important -- there is no spurious barrier to the entry of new suppliers. In a free market, theory says that prices will adjust to be the cost of production plus a profit that is about just enough to make that market as profitable as any other activity.
The opposite of a free market is an oligopoly (including monopoly), where there are few suppliers and new ones are prevented from entry (even if they have the capital and capability to do so). Then the suppliers can conspire to raise their prices to the level that maximizes their net revenue, which can be much higher than the free market price.
Left to themselves, markets often degenerate into oligopolies or monopolies, because of the same factors that led to concentration of bitcoin mining. Many countries have antitrust and competition laws to prevent that from happening and keep the markets free.