Is the aforementioned marriage Islamically valid, given the doubts surrounding it, the family and psychological problems, in addition to the coldness of the marital relationship and the high frequency of divorce between the spouses?

What you mentioned about that woman contained many errors. Your relationship with her before marriage was a sin, and your marriage to her without a guardian (wali) or witnesses is also a sin. You must sincerely repent. This marriage is an invalid marriage and must be annulled. However, it entails the effects of a valid marriage, such as the attribution of the child to you and the necessity of divorce. Divorce in an invalid marriage is irrevocable (bain), so if you took her back without a new contract, the return is void, and she is not your wife. The children are attributed to you, as Muslims have agreed that in any marriage where the husband believed it to be permissible, his child is attributed to him and they inherit from each other. We advise raising the issue with the judiciary or verbally consulting scholars about it, and staying away from that woman until a legally valid marriage contract, fulfilling all conditions, is concluded.