What is the ruling of Islamic law regarding the aforementioned marriages and divorces, and is this wife permissible for the husband after all these divorces, knowing that she has three children from him?

Marriage according to the Russian method is not considered valid due to the absence of a Muslim guardian (wali) for the woman. As for the second marriage, it is valid if the woman entrusted her affair to a Muslim to contract it for her in the presence of two just witnesses.

The third divorce occurred with your solitary travel; it is a conditional divorce. And your statement, "The fourth time is an explicit, enforceable divorce," completed the three divorces. Thus, your wife is not permissible for you until she marries another husband.

The children are your children; you are obligated to raise them, spend on them, and keep them with you. You are not obligated to anything towards their mother because she has become a stranger to you.

It is advised to repent from adultery and choose a righteous wife.