Basic Points
The important things to note with all of these questions is that:
1) We free-market anarchists (FMA) do not claim the magical ability to predict the future that statists of various persuasions claim. We can only offer solutions based on logic, reason, observation, and historical examination. There is no single FMA society; individual societies would compete with each other in numerous ways with the most successful forcing, via market and other processes, the less successful to change or be absorbed.
2) There would probably be a multitude of FMA societies, each with its own quirks.
3) There would probably be a slow drift towards standardization of various rules and customs between FMA societies.
4) FMAs don't believe in the possibility of nor strive for a utopia. We recognize there will always be problems and seek methods to first minimize, then isolate, and finally correct them.
5) There must be a distinction drawn between those who are attracted to libertarianism and FMA because of the underlying theories, and those merely attracted to them because they don't want to be told what to do or are in reality nihilists or apathetic. Societies which claim the banner of FMA but act in a manner closer to nihilism or apathy would not last long; they would fall due to many factors including a lack of coordinated response to external aggression and ineffective economic operations.
6) In a similar way to Marxism, the establishment and continuation of a FMA society requires a different sort of individual than we see in the majority of Western societies today. However, unlike Marxism we don't need to create a new man, but to revert man to the "moral" state which existed prior to modernism and postmodernism. Such a reversion cannot be by force but via the peaceful means of the free-market, logic, and reason. Reversion using force would be like trying to put out a fire using natural gas.
7) Contrary to what individuals such as Jeffrey Tucker and others appear to believe, that humans are interchangeable, the truth is that they are not; culture, language, traditions, work ethic, and religion are just a few of numerous items which can prevent a society from being a cohesive unit. A cohesive society is a prerequisite for a stable society. Such cohesiveness comes not from government schooling, forced association, and so on but arises organically.
8) Although a FMA "society" may function in a similar manner to a state with respect to entities outside of the society, is not a state. It is a collection of individuals, companies, charities, and so on which may choose to enter and leave said society at any time, provided that they have not violated the NAP and as a result have claims against their person. An individual may be a member of more than one society or group at a time; for example they could be a member of their local property owners' association which establishes rules for the use of the property on which their home is located, while also being a member of an engineering or medical society which establishes qualification and other standards for the practice of a profession. The latter is not to give permission to engage in a profession, but to serve as an insurance and/or rating agency.
9) The Non-Aggression Principle (NAP) is not some form of Holy Grail, it is merely a guide which aids in determining whether actions are "moral" or not. Many things within a group and society would be considered under societal norms. For example, in one society it may be acceptable to use vulgar language in from of children, while in another it would be unacceptable. Retaliation in a FMA society does not necessarily equal the initiation of force, it could be as simple as refusing to associate in any way with an individual who is the target of retaliation.
Q & A
The country will probably be vulnerable during the transition and right after, how do you prevent invasions or other kinds of aggression from other countries?
Invasion is prevented during a transition from state to FMA society by first removing prohibitions on firearms ownership while simultaneously removing all penalties for using firearms to defend yourself, your property, and those around you from the initiation of physical force. This would result in an increase in both the number and types of privately held firearms and the number of individuals who carry firearms with them.
Additionally, it is essential that the state immediately relinquish control of the currency and strike any legislation establishing a "legal tender". This is important because controlling a population becomes increasingly difficult if the government ruling said population has neither legal tender laws nor a state currency. By doing this it becomes difficult and unprofitable for a foreign government to take over the state during its transition.
It is at this point that the national government would relinquish most of its functions to more local governments. These local governments would in-turn relinquish their functions to governments even closer to their citizens and smaller in size. This relinquishing of functions would quickly reach the city level, at which point individual neighborhoods could set up their own homeowners' associations through which they establish rules and standards to live within them.
How does a transition from a welfare state to no state happen?
This would occur at the national level by pushing welfare and other HHS programs to local governments, charities, and other private concerns. International and "corporate" welfare would cease at the national level because programs enabling these forms of welfare wouldn't exist.
How does the country deal with immigration and how would you prevent non-violent invasion trough immigration?
I assume "the country" means a state during transition from a state to a number of FMA societies. In this case you wouldn't permit immigration. During such a transition society may experience radical changes and thus be weak, and immigration would likely further weaken it rather than be a benefit. However, as the state weakens and FMA societies strengthen, each would reach the point where they would control their own immigration policies.
If "the country" refers to a FMA society, immigration could be controlled via property owners' associations in a given geographic area. It should go without saying that individuals would have the ultimate decision over who is permitted on their property, given that no previous contracts they entered into place prohibitions on the property.
The nature of a FMA society would make living in it self-selective, naturally selecting those who wish to become part of, or integrate into, the FMA society. The current issues with invasion via immigration only exist because governments enable immigrants to enter into their countries and then refuse to integrate via welfare and other social programs. The market is a powerful driver for integration. As a last-resort, the nature of a FMA society is also one which quickly responds to invasion via immigration; when individuals and groups feel they are being invaded they will respond by cutting off immigration or refusing to associate with immigrants. As a result, immigrants will find it difficult to lead a decent life and would likely self-deport.
What about diplomacy? How exactly would other countries deal with us?
Societies and other groups could assign third-parties to conduct diplomacy on their behalf, or as individuals if a state is willing to engage in diplomacy on the individual level.
What about extradition of those wanted for breaking laws of their respective countries?
This depends on diplomatic relations, the nature of laws claimed to have been broken, and the punishment if found guilty among other items. Most FMA societies would probably surrender citizens of states to their respective states if proof was shown that the individual violated the NAP. It would be in the interest of FMA societies to do their best to maintain good relations with states.
How could an individual protect himself if he does not have a contract with a Private Defense Agency (PDA)?
First, by not engaging in activities or business practices which would result in the need for a PDA. Second, by establishing an excellent reputation in their society and excellent rapport with those they routinely interact with. Third, establish and maintain close friendships with those who they live near. Fourth, by maintaining firearms, skills to use firearms, and skills in physical combat such as Krav Maga.
What if I, as a parent violate the NAP and do harm to my children, who protects them?
As stated before, the NAP is not a "I don't care as long as you don't harm me", but a universal set of concepts which are applied across a society. Therefore the correct thought concerning force and the NAP is "I don't care as long as you don't initiate force on anyone". Violations of the NAP would be handled differently by each society, but it is probable that harming children would be met with some form of force, even if only to remove the child from the abusive parent. Whether this is accomplished by family, neighbors, or PDAs depends on the society, situation, severity of abuse, and threat to life among other things.
The reason for this is that harming children results in mental damage to the child, and to a lesser extent the adult, which in-turn paves the way for behavior issues later in life which may affect the society.
Are there any "basic" rights that everyone would have?
All humans have basic Rights. No human (Individual A) has the "right" to be given products or services at cost to another human (Individual B), whether such cost is in the present or future, unless Individual B has first agreed to provide such services explicitly via contract or implicitly via action.
At what age does one gain, in a sense, "full rights" or do you have them at birth?
"Full Rights" are something an individual is born with; however, individuals are not born with full moral agency nor the ability to comprehend and choose to abide by the NAP or any other set of ideas. It is only with comprehension of the NAP that full moral agency may be attained. As an electrical and software engineer, I take a programmer's view of Rights.
Humans are a class of living organism. This class is derived from, and only exists due to, the class known as infants. To take this further, the class infants only exists due to the class of "complete set of human DNA", ie a developing fetus. A fetus will eventually develop into an infant, which will in-turn develop into an adult capable of comprehending and choosing to abide by the NAP. Therefore, all humans are born with the same inherent Rights.
In contrast, a dog or cat is not of a class whose members are capable, at any point in their life cycle, of understanding and choosing the NAP.
If my child decides to run away and I try to stop him I would be violating the NAP?
It depends on the view your society takes on the parent-child relationship with regards to the NAP. It's likely that if you are considered the child's guardian that this wouldn't be a violation of the NAP.
If I am unable or refuse to provide my child with PDA services, who does?
If you are unable to you could seek assistance from charities. The chances are that market competition would solve the problem on its own. The case of you refusing to provide your child with PDA services depends on the view of the parent-child relationship as well as whether PDAs are the method by which your society prefers to offer such services.
Presuming I have the necessary resources and some mental issues, what would stop me from building a nuke and/or using it? What if I use it on another country who has a state, would I be the only one responsible?
I don't find it probable that you would have a desire to build a nuclear weapon if in a FMA society, nor do I think it probable that you would be able to use it without being stopped by those in your group or society, by force if need be. That is, unless your FMA society is a nihilist or apathetic one.
What would be my status in other countries?
Your status in a particular state would depend on what diplomatic agreements you, your group or society, or your PDA has established with said state.
What would be the the definition of a "person"?
The definition of "person" would depend on the particular anarchist society, but would likely result in adults being considered "persons"; in the long term it would likely result in all members of the human species, unborn and born, being considered "persons".
What about abortion?
The issue of abortion typically falls into one of two categories; that it is a violation of the NAP and that it is not. Abortion is likely one line along which individual FMA societies would be divided and thus organized. Some would make abortion within their properties an action which results in some form of retribution, even if such retribution is only cutting of all social and economic contact with an individual having the abortion. Others would take no such actions. Still others would treat someone they know had an abortion outside of their society the same as someone who had one within their society.
I believe that societies would drift towards abortion being a violation of the NAP.
Specifically, if a girl was born to a family who has a contract with a PDA making abortion a punishable action, would she be breaking the NAP by getting an abortion?
Again this depends on your view of the NAP. In this case the NAP doesn't matter because there is an existing contract which stipulates that abortion is a punishable action. The very existence of this contract tells us that this society is probably one which views abortion as a violation of the NAP anyway.
What if she could but was blackmailed by the family to not do it or else she would be kicked out of the house?
This depends on how the society views the relationship between parents and children. Some societies would have no problem with parents kicking a girl out who had an abortion while others would.
I think societies would drift towards the view that parents kicking those they have guardianship over out of the house as a violation of societal norms at the very least, and perhaps a violation of the NAP.
Assuming a contract with a PDA doesn't prohibit me, what would stop me from kicking out my children? At what age could I kick them out?
Nothing would stop you per se; however, the society you live in may take measures to prevent you from doing so and/or retaliate against you.
What would prevent PDA's from forming monopolies and becoming in a sense "mafias" or a mini-state?
Using force is more expensive than trying to conduct your business peacefully. Monopolies and cartels do not long exist without a state to back them. The likelihood of PDAs becoming monopolies isn't high in a free-market.
What happens when certain immoral options become cheaper than the alternative?
As was stated before, FMAs don't claim to have the solution which leads to a utopia. It is likely that such options would occasionally present themselves; however, it is also likely that such options would be short-lived for many reasons, namely supply and demand as well as peer-pressure and being cast out as a ne'er-do-well.