What is the ruling regarding a man who said to his wife, "You are forbidden to me like my sister" until her condition improves, then said to her, "If you leave, you are divorced," while he does not intend divorce, and is expiation required in this case?

If you say to your wife, "You are forbidden like my sister," and you do not intend ẓihār (a pronouncement that makes one's wife unlawful), then there is nothing upon you except that you must offer an expiation for a broken oath (kaffārat yamīn). However, if you intended it as a temporary ẓihār until your wife's situation improves, then if you have intercourse with her before that, you are obliged to offer an expiation for ẓihār.

Regarding your statement, "If you leave, then you are divorced," if your wife does leave, the divorce takes effect according to the majority of scholars. If you only intended it as a threat, then you are obliged to offer an expiation for a broken oath.

It is not permissible for you to strike your wife on her face.

In the event that a divorce occurs, you can take your wife back (revoke the divorce) if you have not divorced her three times. However, if this divorce is the third, she does not become lawful for you until she marries another man out of genuine desire, and then he divorces her.

You must be cautious about swearing by divorce.