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, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
A marriage that lacks the condition of Wali is Faasid (incorrect) according to the opinion of the majority of scholars. It is the opinion that we adopt here in Islamweb and it is substantiated by Sharee'ah evidence.
Being a divorced woman has no effect in this regard because there is no difference in stipulating the Wali in the case of a virgin and a previously-married woman. The previously-married woman differs from the virgin only in that when she is asked about her marriage, she can express her opinion verbally that she accepts that suitor or not. As to the virgin, her silence is sufficient to indicate her acceptance. The Prophet, sallallaahu ‘alayhi wa sallam, said: "The previously-married woman expresses herself, while the virgin’s acceptance is by keeping silent.” [Ahmad]
In another narration, he, sallallaahu ‘alayhi wa sallam, said: "The previously-married woman expresses herself with her tongue."
As for the issue you have stated, there is also a disagreement of scholars about it. However, the opinion we adopt here in Islamweb is what Ibn Qudaamah stated in Al-Mughni that separation in a Faasid marriage necessitates divorce or annulling the marriage. Accordingly, it is not necessary that the young man divorces that woman; as there is nothing wrong with her trying to convince him to give her Khul' (divorce at the request of the wife who pays a compensation). If he refuses to divorce or accept Khul’, then she may refer the case to the Islamic judge or whoever is acting on his behalf such as the Islamic centers in the western countries to annul the marriage. This marriage, even if it is not officially documented, is established by the testimony of the witnesses. For more benefit, please refer to Fataawa 143529 and 196797.
What you have stated regarding Ibn Qudaamah's quote in Al-Mughni and Nizaam's quote in Al-Fataawa Al-Hindiyyah imply no contradiction; rather, they agree with each other. To prove this, it was stated in Al-Fataawa Al-Hindiyyah: "If the marriage is Faasid, the judge is to separate between the man and the woman…" Thus, the author only wanted to explain the effect of that separation on the ‘Iddah (waiting period) and that meaning is clear in his words.
For further benefit, please refer to Fataawa 86773, 86301and 88375.
Allaah Knows best.