If “definition” means “interpretation”, there is no monopoly needed for it. With one basic law it’s not so much the definition as the fact-finding, therefore there can be a private method of resolution of differences/disputes. It helps to have familiarity with how the courts work to understand what I’m talking about here.
If there is only the one basic law - nobody has the right to initiate the use of force - then it follows that what has to be decided in every dispute is who is in the wrong, and what the consequences should be. Working out how to resolve disputes is not necessarily something that has to be done by a quote unquote government official.
The legal monopoly on the use of retaliatory force should be governed by a strict set of rules, so in THAT sense my answer to Q. 1 would be yes - there should be a monopoly on definition.
“Interpretation” is something that is not a monopoly of government/courts now, so why should it be in a proper civilization?
You asked if all citizens are required to pay for it, and I said no. Wanna know why? Because to require everyone to pay for protection is to initiate force. It has to be left up to the individuals to decide when and if they need to pay for protection. Is it in people’s interest to pay for protection? Yes it is, but a strict adherence to reason and rationality means that those who wish to be employed providing defense services must persuade the public to pay for same. That is what I mean by the statement that the government cannot have the right to coerce the public into paying for it. People have to be free to decide for themselves if and when such expenses are necessary and how much. Of course the system would not be unfunded; it would be voluntarily funded by those who value the services.
Want some examples? One of my favorites is contract insurance, which is basically insurance to cover the costs of having to resort to some kind of dispute resolution services. Such insurance could be needed if one party reneges on a contract. Now, if you did not pay the insurance, you could still sue the reneging party, but you’d have to find the money to pay for it out of your own pocket, instead of having insurance pay for it. Of course, the more litigious you are, the higher the premium you will probably have to pay (unless you are successful at litigation, but that’s an issue to sort out with your insurance provider.) Contract insurance is something the government could run, or run in tandem with insurance companies, which would be set up so as to generate funds to keep the courts running continuously and also to generate profit for the insurance company. Claims on policies to launch lawsuits would cause funds to be made available to pay for those lawsuits. In order to launch a lawsuit, either you need insurance or you need your own funds or you need to convince a lawyer to take your case on a contingency basis.
All citizens would be subject to the Law that prohibits the initiation of force. Thus, the ones who disagree either accept the consequences of initiating force, or more likely they can just go live in one of the other 150 countries where force is still permitted.
Let’s face it - this new country is going to be the only one of its kind. It may well need a budget to provide for its own defense against bully nations which are unhappy with seeing this new free country. Paying for that defense is also something that has to be voluntarily done.
People who agree with the system are more likely to pay something for it, because they value their freedom and they value having a means to deal with those who don’t. The citizens who decide they have the right to initiate force are precisely the reason there needs to be a law against initiation of force in the first place. That doesn’t mean that people have the right to initiate force!!! It does mean that people have the right to retaliatory force.