What is the ruling on marrying a woman who is divorced kitābiyan (by written divorce) before the expiration of her iddah, and what should be done after seven months of this marriage?

It appears that the marriage contract is invalid because the woman was still observing her iddah (waiting period) before its expiration. The period you mentioned, during which the woman was separated from her husband, does not guarantee that they did not meet. She must observe an iddah of three quru' (menstrual cycles) after the divorce, because the iddah is a right of Allah, and disbelievers are addressed by the ancillary rulings of Sharia. What you did is not correct. You must now abstain from her until she observes istibra' (cleansing the womb) with three quru', then you can perform a new marriage contract with her. If she becomes pregnant by you during this period, the pregnancy is attributed to you, because your intercourse in this situation averts the hadd (prescribed punishment), and anything that averts the hadd also attributes the child.