Is the aforementioned 'Urfi' marriage considered Islamically valid, and if not, is the wife entitled to refuse to return after being sent back following the divorce, and is she still considered married to the husband or is she considered divorced?

The majority of scholars hold that marriage without a guardian (wali) is invalid and must be annulled. However, Abu Hanifa considers it valid and does not deem it adultery leading to the prescribed punishment (hadd), because punishments are averted by doubts. If a divorce occurs based on the opinion of the majority, it is an irrevocable divorce (طلاق بائن), and the woman has the right to refuse reconciliation. As for the opinion of Abu Hanifa, the husband may reconcile with her before the waiting period (iddah) ends, and she does not have the right to refuse. After the waiting period expires, reconciliation is not valid except through a new contract with its pillars (a guardian, two just witnesses, and a dowry). The waiting period ends after three menstrual cycles, or three months for a woman who does not menstruate, or upon delivery for a pregnant woman.