Doesn't make sense to have private courts since anybody with a large enough power (e.g. hired thugs) can take them over.
If you mean that the individuals in such a scheme would agree to pay each to have a "private court/police" enforce their contracts etc, then that's the same as government, just one without voting (and presumably there would be some way to settle how those courts/police would operate too, which might end up as a contract itself, but people will have to agree to its terms first, so there's some voting there too).
>If the government helps a company it's definitely not pure capitalism regardless of what the exact definition is.
How that makes it anything more than wishful thinking though? If the government officials pocket money to favor one company, that's fully compatible with the greed and egoism that's essential for capitalism (according to Adam Smith and co “It is not from the benevolence of the butcher, the brewer, or the baker that we expect our dinner, but from their regard to their own interest").
What would prevent them to do it under the table, aside from morals?