Is the marriage that took place between the questioner and the aforementioned girl Islamically valid? If not, is it considered adultery, and are the children in this case considered children of adultery? What must the questioner do to please Allah?

If the one who married the girl to you was a just Muslim, and two Muslim witnesses attested to the contract, and after her iddah (waiting period) from her first husband concluded, then your marriage is valid. However, if she married herself to you without a guardian (wali), the marriage is invalid according to the majority of scholars. Yet, this is not considered zina (fornication/adultery), and the children are attributed to you. In that case, you must then contract a new marriage with her, with a Muslim guardian and two witnesses.