What is the ruling on someone who swore by divorce to his wife twice? Does the first divorce take effect if he has intercourse with her and pays expiation? And what is the expiation for the second divorce?

The majority of scholars hold that a conditional divorce takes effect upon the fulfillment of the condition, and this is the preponderant view. If you made your wife’s divorce conditional upon sexual intercourse and then performed it, the divorce has occurred according to the majority of scholars, and expiation for an oath (kaffarat yamin (definition)) is not sufficient. However, if you did not intend divorce, then according to Sheikh al-Islam Ibn Taymiyyah, expiation for an oath is required.

Regarding your statement: "And now I swore divorce upon her to her face," if you divorced her face to face, this is an effective divorce. This second divorce is binding if your wife was still under your legal guardianship (in your 'isma) during the waiting period ('idda) of the first divorce, or if you had taken her back before the 'idda ended. However, if the second divorce occurred after her 'idda ended without reconciliation, it is not effective.

In the event that one or two divorces are binding, you can reconcile with your wife before her 'idda ends, provided the total number of divorces has not reached three. If her 'idda expires without reconciliation, she does not become lawful for you except through a new marriage contract.