Is the aforementioned marriage considered Islamically valid, is the child born from it legitimate, and are the divorces that occurred during this marriage counted?

What actually took place was not a legally valid marriage, due to the absence of the formula, the guardian (wali), and witnesses. Therefore, the divorce that occurred within it holds no validity. This child is your son and is attributed to you. You must separate from this woman. If you wish to marry her, the contract must be renewed in a proper manner. It is strange that a marriage would be conducted in this way in a Muslim country. What is religiously required of a Muslim is to gain deep understanding of their religion, and whoever is responsible for facilitating marriage must fear Allah Almighty, because the default state for private parts is prohibition, and their permissibility is only allowed through a certain and undeniable path.