This appears to be a difficult situation for you. It is hard for me to comment on the situation itself as I have no information but I can say something about the issue of marriage.
The bible has much to say about marriage. Indeed the imagery of marriage plays an important part in the unfolding story of God’s commitment to us. Starting with Gen 2 where Adam and Eve are from the one flesh and set a precedence for those who follow with instructions to leave their father and mother and be united in ‘one flesh’. This relationship is important to God. It reflects his own relationship within himself (father, Son and Spirit all in relationship with each other) and his relationship with humanity - particularly Israel (c.f. Isaiah 54:1-10 or Jer 3:14). It is also the image that is used to describe Jesus’ relationship with the church (Eph 5:22-33 and Rev 19 & 21).
The key to this relationship is faithfulness. As God is faithful so the husband and wife are to be faithful. God hates divorce (Mal 2:16) and condemns Israel for being unfaithful to him (Jer 3). God’s expectation of marriage is that it is a lifelong singular committment to your partner - no exceptions.
So the short answer to your question is Yes, two people can be considered married from a biblical perspective if they commit themselves to each other before God (indeed the heart of the marriage ceremony is the vows made to one another). I do not know of any stipulation in the bible that says they must do it a particular way nor that there needs to be witnesses. But given that, God will hold them accountable to that commitment, expecting them to be committed to each other for life, through thick and thin and every conceivable temptation. It is a huge commitment and I would never recommend that a couple take it lightly, without reference to family or friends or privately just before God. It is not the sort of commitment to be made in the heat of the moment but rather in the light of day and after some consideration.
After all why would two people who are in love and want to be married want to get married in secret and privately? Marriage is seen as a great cause for celebration and joy. Why would anyone not want to celebrate one of the most important and happy occasions of your life with family and friends? I would be concerned that something more sinister was going on. Perhaps it is a way of getting around the criticisms or prohibitions of family and friends.
Indeed, since it is such a huge commitment for life and since our society is rife with adultery and marriage breakdown, every couple committed to marriage needs all the support and help they can get. Marriage preparation beforehand so they can be aware of all that marriage will involve and then the support and help of their family and friends (particularly their church family and friends) who can keep you accountable and help them overcome temptation and weakness.
Hope this has been helpful, about the other issues if you could post a more specific question relating to each one you will get a reply!