Does the divorce, by which I swore to my wife twice, and which was reviewed before the expiry of the waiting period, take effect, knowing that it was not in the presence of witnesses or a document? And is it permissible to remain with her in the same house after the third divorce, which makes her unlawful [for me], until the completion of her travel procedures?

If by the oath of divorce you intended to divorce your wife, then the divorce occurs even if it is not documented or witnessed. However, if you intended to make the divorce conditional on an event, scholars have differed. The majority of them hold that it is a conditional divorce that takes effect upon the fulfillment of the condition, while Sheikh Al-Islam Ibn Taymiyyah is of the opinion that it is an oath whose expiation is like the expiation for an oath, unless the husband intended the divorce to occur. The view of the majority is the stronger one. If the divorce occurs three times, the wife becomes irrevocably divorced and unlawful to her husband, and it is not permissible for them to live together unless the place is spacious enough that he does not see her awrah. It is advisable to proceed immediately with the divorce documentation procedures to avoid prohibitions.