All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
These acts which you mentioned in the question are called in religious terms Zina (fornication and/or adultery) because they lead to it, and they are strictly forbidden. It is really a great tragedy for this woman to establish a relationship with this man while she is already married; for more benefit, please refer to Fatwa 89681. The worst of all is her statement that the Quran is not suitable for this era, and that she dislikes many rulings in it, and that there is no mercy in Islam. Moreover, she denies the pleasure and the punishment of the dead in their graves. If it is confirmed that she said these statements, then she is an apostate. For more benefit, please refer to Fatwa 82417.
Hence, it is not permissible for her husband to have sexual intercourse with her and he should not allow her to foster his two daughters as long as she persists in her condition. He should endeavour his utmost not to allow her to foster his daughters and he should be keen to nurture them according to religious principles and good moral conduct. If it is necessary to migrate with them from that country, then he should do so. For more benefit, please refer to Fatwa 89418.
Finally, it is an obligation to advise this woman and remind her of Allaah, and seek the help of scholars in that country so that they would repel those misconceptions from her. If she repents to Allaah and embraces Islam during her waiting period, then she may remain with her husband without renewing the marriage contract. However, if her waiting period expires, then a new marriage contract should be conducted; of course if she has repented.
What happened to this woman and what might happen to other women like her, is great evidence proving the seriousness of residing in non-Muslim countries; therefore, one should be very careful about this.
For more benefit, please refer to Fatwa 86405.
Allaah Knows best.