before she had completed the rites of ʿumrah, and you then finished the ʿumrah on her behalf, this is a praiseworthy deed. Your doing so was a precaution; had you not completed it for her, she would still have been considered to have done what she could and would have died while engaged in the act of worship. The Prophet ﷺ, in the incident of the man whose mount threw him at ʿArafah and who died while in a state of ihrām, ordered that he be buried in his ihrām garments, forbade that his head be covered, and forbade that he be perfumed. He said: “He will be raised on the Day of Resurrection proclaiming the talbiyah.” This indicates that if a pilgrim dies while in ihrām, he remains in that state until he is resurrected and nothing of the rites is performed on his behalf, for the Messenger ﷺ did not instruct the Companions to complete the rest of the pilgrimage for that man; he only said, “He will be raised on the Day of Resurrection proclaiming the talbiyah,” meaning that he remains upon his ihrām.
Your mother is most probably included in this ruling, for she died in the course of the ritual and will remain in that state until Allah resurrects her. Nevertheless, since you went on to complete the remaining rites for her, this was done as a precaution—may Allah reward you with goodness.
As for the second part of your question—your having performed ḥajj on her behalf after her death as an act of filial piety and kindness—this is something for which you are to be commended, and we ask Allah to accept from us and from you all righteous deeds.
As for your omission of the saʿy, this is an error, for you are obliged to complete the rites. The saʿy is a pillar of ḥajj, and the pilgrimage is not valid without it. If you entered the state of ihrām