Are divorce proceedings and the rejection of mediation considered a legitimate divorce, or is it a condition that the word "divorce" be uttered? And if the divorce has indeed occurred, can the wife be taken back without a new contract and mahr, and what must he do?

If the husband did not utter an explicit divorce and did not intend divorce by his words, then no divorce is binding upon him. And going to court is not considered a divorce unless he utters it or writes it. If the husband pronounces divorce verbally or in writing, he may take his wife back during her iddah (waiting period) if it was the first or second divorce. The iddah begins from the time of divorce. If the iddah expires without him taking her back, she becomes irrevocably separated from him and is not lawful for him again except with a new contract and dowry. The iddah for a woman is three menstrual cycles, or three months for a woman who has despaired of menstruation or has not yet menstruated, and delivery for a pregnant woman.