Can the legitimate seclusion [khalwa shar'iyya] without consummation, seeking divorce based on illogical reasons, then requesting to conceal the seclusion to avoid the waiting period ['idda], and subsequently becoming engaged during the waiting period, be considered valid legitimate rulings in Islam?

Your question involves the commission of several forbidden acts:

1. Seeking divorce from your husband without a legitimate شرعي excuse, which is Islamically prohibited, unless you were unaware of its prohibition. 2. Your command to your husband to lie and commit forgery, which is a great abomination, and for which your husband also bears sin. 3. If a valid شرعية seclusion (خلوة) occurred, then the waiting period (عدة) may become obligatory according to the majority of scholars, contrary to the new view within the Shafi'i school. 4. The judge's ruling that the divorce is irrevocable (بائن) holds no validity in reality, as it was based on false information; the divorce is revocable (رجعي). If someone proposes to you while you are in your waiting period, the sin is upon you. If a marriage contract takes place before the completion of the waiting period, it is invalid according to the majority of scholars who affirm the obligation of the waiting period due to seclusion, and valid according to the new view of the Shafi'i school.