The first paragraph sounds like:
If nobody cares, than it is ok, to sell your child.
I can't help what you think it sounds like, that's not what I wrote.
Nobody means, nobody in the near neighborhood. So, if my neighborhood doesn't like, when I sell my child, I just can move to a neighborhood, where nobody cares.
That's not what nobody means, but in the situation that everyone that finds out doesn't care, then no one will do anything about it, regardless of the morality of it. That's the same for any ideology in any society everywhere.
Also, what does "then it would be appropriate for anyone to take that child from them and to do a better job" mean?
You mean taking by force? Wouldn't that violate NAP? Can I just take a child from people, when I think, they are shitty parents?
Yes, individuals have the same freedoms as groups, so if it's ok for societal mechanisms to take away children, then it's ok for individuals to take children. However, in an Anarcho-Capitalist society, they would need to deal with any defense agencies that would prevent this sort of thing unless certain criteria were met, like the child was being abused. If the defense agency disagrees, then it would need to be arbitrated by a third party, or there could be violent conflict.
So you can't just take someone's child for any reason you like, it needs to be supported by evidence of violating the Non-Aggression Principal. (Or violating terms of both parties defense agency, but I won't get into that scenario at the moment.)
"Force" does not violate the Non-Aggression Principle. The NAP deals specifically with the initiation of force, not force in general.
http://mises.org/journals/scholar/Hoppe.pdfThe second paragraph seems to tell a whole different story:
Why is it rape?
If it is about fucking up their psyche and health, than I am wondering what is the difference to letting them work in a mine?
Any non-consensual sexual intercourse is rape. There are all sorts of other kinds of sexual activities that wouldn't be rape, but things like oral sex, sexual harassment, and so on would still be violations of the Non-Aggression principal for the same reason; The child cannot provide consent.
Working in a mine does not require consent of the child, it is true that this should be avoided if at all possible, but the reason is not because the child didn't consent. The parent can tell the child to do all sorts of things without the child's consent. For example; "Don't shit in the living room.", "Eat your vegetables or go to bed.", "Go to school." and so on. I'm sure there are gray areas here, but overall the parent has a responsibility to make decisions for the child that are best for the child, and when the parent doesn't do this, they are abusing the child and therefore violating the Non-Aggression Principal.
In some cases, I'm sorry to be the bearer of bad news, putting children to work is the best option for the child. Putting children under the age of consent to work as sex slaves is never a moral thing to do, though if you're extremely creative you could come up with nearly-impossible scenarios in which it just can't be stopped.