What is the New Testament "process" for doing a church plant? Given that new churches are to be started (eg the ministry of the Apostle Paul in partial fulfillment of the Great Commission), what is the New Testament model or process for how this should be done?

Does it vary culturally?

Can (or 'how can') one know when it is time to plant another church in the same town\region, and when to encourage people to come to an established local body?

Most specifically, I'm looking for models\processes surrounding "independent" or "non-denominational" churches - ie not Roman Catholic, Episcopalian, PCA, etc who all have formal processes for starting new churches.
Many independent churches first sprang up after the partial collapse of Catholicism during the reformation possibly using the very same New Testament guidelines and principles you seek. The most illuminating case know of that sheds light on the underpinning of New Testament principles at this time is the arguments between Anglican ministers and the independents which has been partially recorded in Johns Owens ‘A Brief Vindication of the Nonconformists from the Charge of Schism’, in Owen’s Works Volume 13.

The situation is very interesting because there was a vacuum created by so many exiting the Catholic church and the Pope’s loss of powers with various princes and Kings, that new or strengthened ‘state approved’ churches formed, or otherwise tried to extend their influence and power. However, as in the case of England, where Anglicanism was by no means newly formed, the established church with its ‘traditions’ wanted every Protestant in its ‘control’ and tried to repress the existence of independent churches. It was under these conditions that John Owen made his response to the Anglican accusation of unlawful and sinful schisms within the Church of England.
From the independent revolt of the Church of England you get the following New Testament guidelines that the independents argued: 

1-State approved Churches carry no more authority than Christ approved churches:
The New Testament nowhere suggests that the church needs approval from government officials, therefore anyone trying to fuse church and state has no solid basis to present an argument apart from its claimed traditions.

2-Rather than external tradition the New testament established a rule of love and forbearance with one another who hold differing views on non-essential doctrines and practices: Those claiming the authority of tradition claim it was first handed down by the Apostles but if this was true it would be clearly defined in scripture due to its importance. In reality the Apostles formulated rules of ‘brother love’ while many traditions and ceremonies of the Jews were brought to a slow and graceful death. As the new church was a mixture of Jew and Gentile with some Jews having a weaker conscience concerning some matters such as eating meats sacrificed to idols, the Apostles asked those with free consciences to lovingly submit to meaningless rules so as not to offend their brother. (1 Cor. 8:13). However when a wicked Judaizer insisted that the free follow a particular tradition as necessary for living a godly life, even when the Apostles have already said it was not necessary, then Christians absolutely refused as it became no longer a harmless difference but an attack on the gospel. (Acts 16:3, Gal 2:3, 5:2). These principles can be used when any denomination tries to repress the formation of a church upon grounds of the tradition they hold, for the repression itself on these grounds constitutes a possible good reason for a separation.

3-The New Testament, shows that churches naturally sprang up wherever new groups of believers were formed, sometimes starting in peoples homes, and were never pressured to join under subjection to a ‘higher’ church or central authority. As these churches later fell into various forms of Apostasy they established their own conflicting traditions, not one universal tradition under an external rule, until later imposed by the Roman government. This free-forming tradition can be seen by the early division over whose tradition should be followed concerning Easter, as mentioned in this post.

4 The ideal church in not represented as a ‘mega-church’ or just a few members of a family but a reasonably sized group who by virtue of their differing gifts and in some matters differing ideas, are most suitable for the edification and strengthening in the faith of each member. (1 Corinthians 12:18-27). The reason why a family by itself is not a proper church is that members must be freely willing to join and it would not have a wide number of differing gifts unless that family was particularly large. The idea of a mega-church does not seem to suit the ideal New Testament model, for although 5000 were saved in one day, we have no records of a church that big assembling in the New Testament, rather churches sprouted up all over  like a natural vines growing around every natural turn and bend in the existing communities. (I am not saying churches can’t be huge or tiny, only that it does not seem ideal. Of course a mega-church can be divided into many smaller churches that just happen to meet in the same building). This sort of creates a flexible rule that if a church is made for believers (not believers made for churches) a church should bend its structure to best support the growth of believers. If a church is so large that intimate fellowship and properly organized teaching and administration becomes overly difficult, it should reasonably consider forming a new church (whether through subdividing itself, or finding a new location more suitable to many of its members).

5 A new church is not meant as a means to simply collect those who are merely argumentative, proud, or otherwise divisive over disputable matters. (Rom 14:1).  A great deal of Philippians’ seems to be related to having humility and striving towards the goal (Phil 3:12-17, 1 Cor 13:12) which includes a goal of a better understanding. While dodging those non-neutral attacks of our faith (Phil 3:2). We also must avoid useless divisions through modelling humility and love (Phil 4, Phil 2).  Since we have never arrived at a perfect understanding, we must allow the fact that everyone is at a different stage of progress and coming from different starting points, so we must chose to serve in such a way as to help each ‘child’ grow and ‘woe to him’ who causing the child to stumble. (Mark 9:42). We ourselves must be like children, not wondering how will be the greatest. 

6  There may comes a point where a sincere believer or entire body can no longer in good conscience remain within a certain fellowship where disputable matters are no longer allowed to be left in love and humility, are enforced against the conscience of others. When an unnecessary saw of constant discord is swung back and forth between believers by argumentative leaders, or other situations arise, making it seem impossible to properly advance in one’s knowledge in Christian love and humility, believers may need to form new churches or join other ones. This has sometimes been the cause of why one party ‘breaks away’ from the traditions of another, because those holding the tradition become aggressive and seek to make everybody rankled over argumentative subjects without any pause for relief. This ungodly spirit of argumentation may even occur on this site, when we seek to only argue with a saw of contention (rather then passionately disagree) or seize upon ‘another’s expressions’ in a way that does not encourage their intended meaning, offering no support in humbly drawing it out to the edification of a very similar faith. It is best to avoid contact with those in such a state of mind for our own safety.

7 A church is meant to be established the same simple structure of New Testament church without additional layers of authority invented by men or tradition. This can be deduced from the fact that no such layers like ‘arch-teacher’, or ‘lord-deacon’, or whatever title might be created are not present in the New testament which opposes the spirit of humility and love as the true rule the Apostles established for church government. This means the roles of deacons or elders, as well as overseer, teacher and pastor must be established and then formed in the new church on order to have orderly worship as well as ensuring the observance of the Lord’s Supper and Baptism are administered. If there are not sufficiently qualified people with the necessary strengths to effectively support a healthy development of a believing body, this should not be attempted unless there are no other such churches within reasonable distance form ones own community.

I do not know if this answers your question as it is an unusually large question with many possible aspects to a solid answer, but this is all I can manage to collect.