Yes, Locke believes that these things should exist. He does not say or believe that they will exist or be respected in a state of nature--see "ought". For enforcement, he turns to government. I'm just puzzled as to why you bothered to bring him up as if he was in support of non-government or consistency with the NAP. He clearly was not.
I brought him up because he was somebody who thought that government was only justified if it rested on consent (and anarchists say it does not and cannot), and so is a useful example in explaining
why we talk about consent being needed. You said that using violence against a criminal, say a murderer, who had not consented to being bound by our laws was an initiation of force, but the fact is that consent theorists, like Locke, and like we anarchists, think consent is necessary precisely
because we think people are all bound by enforcible principle's of justice whether we like it or not, and it is consequent to this that there are things that, if we do them to other people without their consent, are in violation of those principles of justice. Likewise, there are things that, if we do them to or with various objects without consent or permission from specific others, we violate those principles of justice.
So, neither Locke, nor any libertarian, anarchists included, thinks you need permission from would-be murderers in order to use violence to prevent or punish that guy's actions.
I'm with you on point #1. I'm fine with resolving disputes in private courts. In fact, it happens all the time.
I disagree with point #2. It is possible to be impartial when you are the complaining party, and I would say it is necessary in the case of the murder of a person with no heirs and nobody with any legitimate claim to the person's estate, etc.
It is possible, but it is surely far from probable, is it? I mean, if we said that in every murder case the person deciding whether or not a crime has been committed should be the accused, nobody would ever get any justice, would they? It would be
possible for a guilty murderer to be impartial and decide, yes, they are guilty, but not likely, surely. Likewise, it is possible then when I accuse the government of a crime, it could impartially decide, yes it did commit a crime, but not likely. And I don't think you would want to leave it up to a person you suspected had wronged you to decide whether he had or not.
I would like to bring up a point #3: I believe that property rights of individuals should be enforced (pursuant to their wishes, if specified) regardless of whether or not a complaining party exists after a crime has been committed. I see no way to guarantee this in a completely private system.
This relates to your point about being murdered without leaving an heir, or someone with a legitimate claim to the estate, right? Well, firstly, I can set up a contract with a protection agency that, in the event of my murder, it does its upmost to investigate the crime, and punish the wrongdoer. Firms that were paid for such services but failed to do so would lose reputation and tend to lose business to those that kept such contracts.
Secondly, this sounds like a good incentive to ensure I do have an heir or someone to take over the estate, or to leave my claims against wrongdoers to.
An alternative is that if somebody dies leaving no heir and with nobody having a legitimate claim to the estate, then that estate becomes effectively unowned property, and maybe homesteaded. And, since claims against wrongdoers are the property of their victim, that means that claim can be homesteaded and enforced by the wrongdoer.
Honestly, laws have been privately enforced for longer than they publicly enforced. I imagine that they found someway to resolve this issue.