What is the ruling regarding a couple returning to each other after two divorces and the filing of a third divorce case, before it is judicially settled, knowing that sexual intercourse occurred between them during the period of separation?

If the divorce claim was merely a formal procedure and no actual divorce was issued by you, then no divorce has taken place, and there is no sin upon either of you for what you did.

However, if you have indeed divorced your wife three times, and then wished to formalize that, then the third divorce has occurred, and your wife becomes unlawful for you until she marries another man and he consummates the marriage with her, and then he divorces her and her waiting period ends. At that point, it becomes permissible for you to marry her again.

Allah Almighty says: "Divorce is twice. Then, either keep [her] in kindness or [send her away] with good treatment."

And Allah Almighty says: "And if he has divorced her [for the third time], then she is not lawful to him afterward until she marries another husband. And if he [the latter husband] divorces her, there is no blame upon them [the former husband and wife] for returning to each other if they think that they can keep [within] the limits of Allah. These are the limits of Allah, which He makes clear to a people who know."

Accordingly, your wife is considered a stranger to you, and it is not permissible for you both to remain together, and you both must repent from that.